Latest revision as of 14:38, 7 October 2016

Abstract

In this paper a standard numerical method with piecewise linear interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problem for second order ordinary delay differential equations with discontinuous convection coefficient and source term. An error estimate is derived by using the supremum norm and it is of almost first order convergence. Numerical results are provided to illustrate the theoretical results.

Keywords

Singularly perturbed problem; Convection–diffusion problem; Discontinuous convection coefficient; Shishkin mesh; Delay

2010 Mathematics Subject Classification

65L10; 65L11; 65L12

1. Introduction

Singularly perturbed ordinary differential equations with a delay are ordinary differential equations in which the highest derivative is multiplied by a small parameter and involving at least one delay term. Such type of equations arises frequently from the mathematical modelling of various practical phenomena, for example, in the modelling of the human pupil-light reflex  [14], the study of bistable devices  [4] and variational problems in control theory  [10], etc. It is important to develop suitable numerical methods to solve singularly perturbed differential equations with a delay, whose accuracy does not depend on the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon } , that is the methods are uniformly convergent with respect to the parameter.

In the past, only very few people had worked in the area Numerical Methods to Singularly Perturbed Delay Differential Equation(SPDDE). But in the recent years, there has been growing interest in this area. The authors of  [12], [6], [16], [1] and [2] suggested some numerical methods for singularly perturbed delay differential equations with continuous data. Recently few authors in  [20], [21] and [17] suggested some numerical method for singularly perturbed delay differential equations with discontinuous data.

In the present paper, as mentioned in the above abstract, motivated by the works of [7], [3] and [13], we consider the following singularly perturbed boundary value problem (2.1) for second order ordinary delay differential equations with discontinuous convection coefficient and suggest a parameter uniform numerical method. It is proved that this method is uniformly convergent of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O(N^{-1}ln^2N)} .

The present paper is organized as follows. In Section  2, the problem of study with discontinuous data is stated. Existence of the solution to the problem is established in Section  3. A maximum principle of the DDE is established in Section  4. Further a stability result is derived. Analytical results of the problem are derived in Section  5. The present numerical method is described in Section  6 and an error estimate is derived in Section  7. Section  8 presents numerical results.

2. Statement of the problem

Through out the paper, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C,C_1}

denote generic positive constants independent of the singular perturbation parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon }
and the discretization parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N}
of the discrete problem. Further, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_N}
denotes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace 0,1,\ldots ,N\rbrace }

. The supremum norm is used for studying the convergence of the numerical solution to the exact solution to a singular perturbation problem: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert u\Vert _{\Omega }=sup_{x\in \Omega }\vert u(x)\vert } .

Motivated by the works of  [8], [3] and [13], we consider the following BVP for SPDDE.

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in Y=C^0(\overline{\Omega })\cap C^1(\Omega )\cap C^2(\Omega ^{{_\ast}})}

such that

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0<\epsilon \ll 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a,f}

are sufficiently smooth and bounded in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^{{_\ast}}}

. The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b}

is a sufficiently smooth function on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\Omega }}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega =(0,2)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\Omega }=[0,2]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^{{_\ast}}=\Omega ^{-}\cup \Omega ^+} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^{-}=(0,1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^+=(1,2)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi }
is smooth on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [-1,0]}

.

The above problem (2.1) is equivalent to

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(1-)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(1+)}
denote the left and right limits of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}
at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=1}

, respectively.

3. Existence result

For the reader’s convenience some known results are briefly reported on this section and in Section  4. They can be used here with some modifications.

Theorem 3.1.

The problem   (2.1)   has a solution  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in C^0(\bar{\Omega })\cap C^1(\Omega )\cap C^2(\Omega ^{{_\ast}})} .

Proof.

The proof is by construction. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_2}
be particular solutions of the DDEs,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_1=\phi (x),x\in [-1,0]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a_1,a_2\in C^2(\bar{\Omega })}

with the above properties.

Consider the function

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _3}
are the solutions of the following problems, respectively:

and

It is easy to see that the above function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y}

satisfies the differential equation  (2.1) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(0)=y(0)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(2)=y(2)}

. Using the similar arguments given in  [7, Theorem 1], and [19, Theorems 2,3] one can prove the existence of the solution. □

Note: For the existence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _i,i=1,2,3}

one may refer to  [18] and [5].

4. Stability result

Theorem 4.1 Maximum Principle.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w\in C^0(\overline{\Omega })\cap C^2(\Omega ^{{_\ast}})} be any function satisfying  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(0)\geq 0} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(2)\geq 0} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Pw(x)\geq 0,\forall x\in \Omega ^{{_\ast}}} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w^{{'}}(1+)-w^{{'}}(1-)=[w^{{'}}](1)\leq 0} . Then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(x)\geq 0,\forall x\in \overline{\Omega }} .

In the following we use the function

Proof.

Using the above function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s}

and the procedure adopted in  [20, Theorem 3.1], one can prove this theorem. □

Corollary 4.2 Stability Result.

For any  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in Y} we have

Proof.

Using the barrier function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \psi ^{\pm }(x)=CC_1s(x)\pm u(x),x\in \overline{\Omega }} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_1=max\lbrace \vert u(0)\vert ,\vert u(2)\vert ,sup_{\xi \in \Omega ^{{_\ast}}}\vert Pu(\xi )\vert \rbrace }

and the procedure adopted in  [20, Theorem 3.2], we can prove this corollary. □

Note: An immediate consequence of the Corollary 4.2 is that, the solution of the BVP (2.1) is unique.

5. Analytical results

Theorem 5.1.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u} be the solution of the problem   (2.1), then we have the following bounds

Proof.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in \Omega ^{-}} . Then we have,

Integrating (2.2) from 0 to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}

we get,

Therefore,

By the mean value theorem there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z\in (0,\epsilon )}

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert \epsilon u^{{'}}(z)\vert \leq 2\Vert u\Vert _{\overline{\Omega }}}

. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon \vert u^{{'}}(0)\vert \leq C(\Vert u\Vert _{\overline{\Omega }}+\Vert f\Vert _{\Omega }+\Vert \phi \Vert _{[-1,0]})} . Hence,

Similarly one can show that, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon \vert u^{{'}}(x)\vert \leq C,x\in \Omega ^+} .

From (2.2) it is easy to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert u^{(k)}\Vert _{\Omega ^{{_\ast}}}\leq C\epsilon ^{-k},k=2,3} . Hence the proof. □

To derive uniform error estimates, we need sharper bounds on the derivatives of the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u} . We derive these using the following decomposition of the solution into smooth and singular components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(x)=v(x)+w(x)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v}
can be written in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v=v_0+\epsilon v_1+\epsilon ^2v_2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_0,v_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_2}
are defined respectively to be the solutions of the following problems:

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_0\in C^0(\Omega ^{{_\ast}})\cap C^1(\Omega ^{{_\ast}})}

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_1\in C^0(\Omega ^{{_\ast}})\cap C^1(\Omega ^{{_\ast}}\cup \lbrace 2\rbrace )}

such that

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_2\in Y^{{_\ast}}}

such that

We assume that, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v_0^{{''}}\Vert _{\Omega ^{{_\ast}}}\leq C}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v_1^{{'''}}\Vert _{\Omega ^{{_\ast}}}\leq C}

.

Thus the smooth component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v}

satisfies the following:

find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v\in C^0(\Omega ^{{_\ast}}\cup \lbrace 0,2\rbrace )\cap C^2(\Omega ^{{_\ast}})}

such that

Further Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w}

satisfies the problem, that is, find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w\in C^0(\Omega ^{{_\ast}}\cup \lbrace 0,2\rbrace )\cap C^2(\Omega ^{{_\ast}})}
such that

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v+w=u\in Y^{{_\ast}}} .

Theorem 5.2.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w} be the solutions of the regular and singular components of the solution  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u} . Then

Proof.

Integrating the differential equation  (5.1)–(5.4) separately on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^{-}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega ^+}

, we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v_i\Vert \leq C,i=0,1}

and by the stability result we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v_2\Vert \leq C}

. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v\Vert _{\Omega ^{{_\ast}}}\leq C} . Similarly one can prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert v^{(k)}\Vert _{\Omega ^{{_\ast}}}\leq C(1+\epsilon ^{2-k}),k=0,1,2,3} .

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w(x)\vert \leq \vert u(x)\vert +\vert v(x)\vert } . From the stability result we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert u(1)\vert \leq C} . Further, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert v(1)\vert \leq C} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w(1)\vert \leq \eta }

(say). Now consider the barrier function

It is easy to check that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi _1^{\pm }(0)=\eta exp(\frac{-\alpha }{\epsilon })\pm w(0)\geq 0,\varphi _1^{\pm }(1)=\eta \pm w(1)\geq 0} .

Applying the result given in  [9, Theorem 2.1] on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]} , we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi _1^{\pm }(x)\geq 0} .

Consider the barrier function

It is easy to see that, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi _2^{\pm }(1)=C_1(\epsilon +1-\epsilon exp(\frac{-\alpha }{\epsilon }))\pm w(1)\geq 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi _2^{\pm }(2)=C_1(\epsilon +exp(\frac{-\alpha }{\epsilon })-\epsilon )\pm w(2)\geq 0} . Again applying the result given in  [9, Theorem 2.1] on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [1,2]} , then we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi _2^{\pm }(x)\geq 0} . Using the procedure adopted in  [7, Lemma 4], one can prove the rest of this theorem. □

Note: From the above theorem it is easy to show that

6. Discrete problem

In this section, mesh selection strategy, namely piecewise uniform mesh (Shishkin mesh), is explained. Also upwind finite difference scheme with piecewise linear interpolation on Shishkin mesh for the problem (2.1) is described.

6.1. Mesh selection strategy

Since the BVP (2.1) exhibits strong interior layers at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=1}

and a weak boundary layer at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=2}

, we choose a piecewise uniform Shishkin mesh on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,2]} . For this we divide the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,2]}

into five subintervals, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _1=[0,1-\tau _1]}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _2=[1-\tau _1,1]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _3=[1,1+\tau _2]} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _4=[1+\tau _2,2-\tau _2]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _5=[2-\tau _2,2]} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _1=min\lbrace 0.5,\frac{2\epsilon lnN}{\alpha }\rbrace ,\tau _2=min\lbrace 0.25,\frac{2\epsilon lnN}{\alpha }\rbrace } . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_1=4N^{-1}(1-\tau _1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_2=4N^{-1}\tau _1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_3=8N^{-1}\tau _2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_4=4N^{-1}(1-2\tau _2)} . The mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\Omega }^N=\lbrace x_0,x_1,\ldots ,x_N\rbrace }

is defined by

6.2. A finite difference scheme for (2.2)

On Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\Omega }^N} , we define the following scheme for the BVP (2.2):

where

6.3. Discrete stability result

Lemma 6.1 Discrete Maximum Principle.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z(x_i)} be a mesh function satisfying  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z(x_0)\geq 0,Z(x_N)\geq 0} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^NZ(x_i)\geq 0,i\in I_N\setminus \lbrace 0,N/2,N\rbrace } and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (D^+-D^{-})Z(x_{N/2})=[DZ](x_{N/2})\leq 0} . Then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z(x_i)\geq 0,\forall x_i\in \overline{\Omega }^N} .

Proof.

Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s(x_i)=\lbrace \begin{array}{c} \frac{3}{2}+\frac{x_i}{2}\mbox{,}\quad x_i\in [0,1]\cap \overline{\Omega }^N\mbox{,}\\ 3-x_i\mbox{,}\quad x_i\in [1,2]\cap \overline{\Omega }^N\mbox{.} \end{array}}

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^Ns(x_i)>0,\forall x_i\in \Omega ^{{_\ast}}\cap \overline{\Omega }^N} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [Ds](x_{N/2})<0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s(x_i)>0,\forall x_i\in \overline{\Omega }^N} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu ^{{_\ast}}=max\lbrace \frac{-Z(x_i)}{s(x_i)}:x_i\in \overline{\Omega }^N\rbrace } . Then there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_i^{{_\ast}}\in \overline{\Omega }^N}

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z(x_i^{{_\ast}})+\mu ^{{_\ast}}s(x_i^{{_\ast}})=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Z(x_i)+\mu ^{{_\ast}}s(x_i)\geq 0,\forall x_i\in \overline{\Omega }^N}

. Therefore the mesh function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (Z+\mu ^{{_\ast}}s)}

attains its minimum at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_i=x_i^{{_\ast}}}

. Suppose the theorem does not hold true, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu ^{{_\ast}}>0} .

Case (i)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle :(x_i^{{_\ast}}\in \Omega ^{-}\cap \overline{\Omega }^N)}

It is a contradiction.

Case (ii)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle :(x_i^{{_\ast}}\in \Omega ^+\cap \overline{\Omega }^N)}

It is a contradiction.

Case (iii)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle :(x_i^{{_\ast}}=x_{N/2})}

It is a contradiction. Hence the proof of the theorem. □

Lemma 6.2.

For any mesh function  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(x_i)} we have

Proof.

One can easily prove this lemma by using Lemma 6.1 and the discrete barrier function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_i)=CC_1s(x_i)\pm U(x_i),x_i\in \overline{\Omega }^N} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle C_1=max\lbrace \vert U(x_0)\vert ,\vert U(x_N)\vert \\\displaystyle max_{j\in I_N\setminus \lbrace 0,N/2,N\rbrace }P^NU(x_j)\rbrace \end{array}} . □

Analogous to the continuous function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u} , we decompose the numerical solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(x_i)}

defined by (6.1)–(6.3) as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(x_i)=V(x_i)+W(x_i)}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V(x_i)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W(x_i)}
satisfy the following:

and

Theorem 6.3.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(x_i)} be the numerical solution of   (2.2)   defined by   (6.1)–(6.3)   and further let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V(x_i)} be the numerical solution of   (5.7)   given by   (6.4). Then,

Proof.

Consider a mesh function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle \varphi ^{\pm }(x_i)=C_1N^{-1}[s(x_i)+\eta (x_i)]+C_1\psi (x_i)\vert U(x_{N/2})-\\\displaystyle -V(x_{N/2})\vert \pm (U(x_i)-V(x_i)),i\in I_N\end{array}}

where

It is easy to see that, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_0)\geq 0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_N)\geq 0}
for a suitable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_1>0}

. Further,

Note that, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\in I_N\setminus \lbrace 0,N/2,N\rbrace } , we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^N(U(x_i)-V(x_i))=0} .

Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^N\varphi ^{\pm }(x_i)\geq 0,i\in I_N\setminus \lbrace 0,N/2,N\rbrace }

by a proper choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_1}

.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_i=x_{N/2}} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle [D]\varphi ^{\pm }(x_i)=-C_1\frac{7N^{-1}}{2}-C_12\vert U(x_{N/2})-\\\displaystyle -V(x_{N/2})\vert \pm [[D]U(x_i)-[D]V(x_i)]\leq 0\end{array}} , by (6.2), (6.4) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle (x_{i+1}-\\\displaystyle -x_{i-1})\vert \delta ^2v(x_i)\vert \leq max_{[x_{i-1},x_{i+1}]}\vert v^{''}(x)\vert N^{-1}\end{array}} [15, page 52]. Then by Lemma 6.1, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_i)\geq 0,\forall i\in I_N} . Hence the proof. □

7. Error analysis

In this section we derive an error estimate for the numerical solution obtained by the scheme (6.1)–(6.3) for the problem (2.1).

Lemma 7.1.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v} be the solution of the problem   (5.7)   and let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V(x_i)} be its numerical solution defined by   (6.4). Then,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert v(x_i)-V(x_i)\vert \leq CN^{-1},i\in I_N} .

Proof.

Now,

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert v^I(x_i)-v(x_i-1)\vert \leq CN^{-2}}

  [11], then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert P^N(v(x_i)-V(x_i))\vert \leq CN^{-1},i\in I_N\setminus \lbrace 0,N/2,N\rbrace }

. Then by Lemma 6.2, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert v(x_i)-V(x_i)\vert \leq CN^{-1},i\in I_N} . Hence the proof. □

Lemma 7.2.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w} be the solution to the problem   (5.8)   and let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W(x_i)} be its numerical solution defined by   (6.5). If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon \leq CN^{-1}} , then we have  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w(x_i)-W(x_i)\vert \leq CN^{-1}ln^2N} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\in I_N} .

Proof.

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w(x_i)-W(x_i)\vert \leq \vert u(x_i)-U(x_i)\vert +\vert v(x_i)-V(x_i)\vert } . Then by Eq. (5.9), Theorem 6.3 and Lemma 7.1, we have

Therefore

Now consider a mesh function

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau =min\lbrace \tau _1,\tau _2\rbrace } . From (7.1), it is clear that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_{N/4})\geq 0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi ^{\pm }(x_{5N/8})\geq 0}
for a suitable choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_1>0}

.

Note that,

Also note that, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w^I(x_i)-w(x_i-1)\vert \leq CN^{-1}}

  [11]. Further, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert P^N(w(x_i)-W(x_i))\vert \leq C_2\epsilon ^{-2}N^{-1}}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_2>0}

a constant independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N}

.

Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^N\varphi ^{\pm }(x_i)\geq 0,i\in \lbrace N/4+1,\ldots ,N/2-1,N/2+1,\ldots ,5N/8-1\rbrace } . Then by the Lemma 6.1, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert w(x_i)-W(x_i)\vert \leq CN^{-1}ln^2N,i=N/4+1,\ldots ,5N/8-1} . Hence the proof. □

Theorem 7.3.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u} be the solution of the problem   (2.2),  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(x_i)} be its numerical solution defined by   (6.1)–(6.3). Then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert u(x_i)-U(x_i)\vert \leq CN^{-1}ln^2N,i\in I_N} .

Proof.

The desired estimate follows from the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u=v+w,U=V+W}

and from the Lemma 7.1 and Lemma 7.2. □

8. Numerical examples

In this section, three examples are given to illustrate the numerical method discussed in this paper. We use the double mesh principle to estimate the error and compute the experiment rate of convergence in our computed solutions for all problems. For this we put Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{\epsilon }^M=max_{0\leq i\leq M}\vert U_i^M-U_{2i}^{2M}\vert } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U_i^M}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U_{2i}^{2M}}
are the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

th components of the numerical solutions on meshes of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2M}
points respectively. We compute the uniform error and rate of convergence as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D^M=max_{\epsilon }D_{\epsilon }^M\mbox{and}\quad p^M=log_2(\frac{D^M}{D^{2M}})}

. For the following examples the numerical results are presented for the values of perturbation parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon \in \lbrace 2^{-27},2^{-12},\cdots ,2^{-6}\rbrace } .

Example 8.1.

Table 1 presents the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D^N}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p^N}
for this problem. Fig. 1 and Fig. 2 represent the numerical solution and the maximum point wise error for this problem, respectively.

(Number of mesh points)

Fig. 1. Numerical solution of the problem stated in Example 8.1.

Fig. 2. Maximum point wise error for the problem stated in Example 8.1.

Example 8.2.

Table 2 presents the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D^N}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p^N}
for this problem. Fig. 3 and Fig. 4 represent the numerical solution and the maximum point wise error for this problem, respectively.

(Number of mesh points)

Fig. 3. Numerical solution of the problem stated in Example 8.2.

Fig. 4. Maximum point wise error for the problem stated in Example 8.2.

Example 8.3.

Table 3 presents the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D^N}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p^N}
for this problem.

(Number of mesh points)

9. Discussion

A BVP for one type of SPDDEs is considered. To obtain an approximate solution to this type of problem, an upwind finite difference scheme with piecewise linear interpolation on Shishkin mesh is presented. The method is shown to be of almost first order convergence. This is very much reflected on the numerical results (Table 1, Table 2 and Table 3). Also Fig. 1 and Fig. 3 represent that the model problems stated in Example 8.1 and Example 8.2 exhibit strong interior layers at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=1}

and a weak boundary layer at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=2}

. Fig. 2 and Fig. 4 represent the maximum point wise error for the numerical solutions. Further these Fig. 2 and Fig. 4 represent the uniform convergence of the numerical method presented in this paper. The authors of  [7] have considered second order ordinary differential equations with discontinuous convection coefficient with different signs on different subdomains. The solution to the problem considered in  [7] exhibits strong interior layers at an interior point. Whereas the problem considered in this paper exhibits strong interior layers at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=1}

and weak boundary layer at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=2}
(see Theorem 5.2). This is due to the presence of the delay term with the differential equation. Therefore, to accommodate these interior layers and boundary layer in numerical solution, the Shishkin mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{\Omega }^N}
has been constructed in Section  6.1. In  [7], the authors have suggested a uniform numerical method without interpolation, whereas the finite difference method with interpolation is needed in this paper, since the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_i-1,i>N/2}
need not be a mesh point.

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