A Numerical Scheme For Semilinear Singularly Perturbed Reaction-Diffusion Problems

Kerem Yamac 1  and Fevzi Erdogan 2
  • 1 Dep. of Mathematics and Science Education, Van
  • 2 Department of Mathematics, Van, Turkey
Kerem Yamac and Fevzi Erdogan

Abstract

In this study we investigated the singularly perturbed boundary value problems for semilinear reaction-difussion equations. We have introduced a basic and computational approach scheme based on Numerov’s type on uniform mesh. We indicated that the method is uniformly convergence, according to the discrete maximum norm, independently of the parameter of ɛ. The proposed method was supported by numerical example.

1 Introduction

Let us take into consideration of the following singularly perturbed semilinear reaction-diffusion boundary value problem:

Lu(x)εu(x)+f(x,u(x))=0,0<x<l,
u(0)=A,u(l)=B.
where ɛ, 0 < ɛ ≤ 1 is the perturbation parameter, f is given sufficiently smooth functions that f (x, u(x)) ∈ C ([0, l]xℝ), f(x,u)uα>0 . The problem (1)(2) has boundary layers at the boundary points.

In a differential equation, if a small parameter is multiplied by the highest- order derivative term in the differential equation, generally it is called the singularly perturbed problem (denoted here by ɛ).[1,2,3]

Second-order reaction-diffusion type boundary value problems with singularly perturbed occur frequently in fluid mechanics and other fields of applied mathematics. Examples of such studied problems can be seen in [4,5,6].

Since the continuous solutions of singularly perturbed problems change very quickly in certain layers, it’s numerical analysis is always important. It is well known that when the parameter of perturbation is small enough, conventional numerical methods to solve the problem do not work well. Therefore should be develop develop appropriate numerical methods for such problems, whose convergence does not depend on the perturbation parameter. It can be found in the literature that there are many numerical finite difference schemes that are stable for all values parameter of perturbation [2,3,4,5,6,7,8,9]. One of the most important ways to easily find the methods that give such results is use of finite difference schemes with exponentially fitted [11,12,13].

As a numerical study, several examples of the second order singularly perturbed convection-diffusion problems can also be seen in [5],[21].

In [1314,16,19] introduced numeric methods and special mesh methods for various reaction-diffusion type problem. In [20] semi-linear reaction diffusion equations are discussed. The discrete and upper of solutions were investigated for the asymptotic properties. And also it is numerical solutions are investigated on pisewise mesh.

The Numerov method is undoubtedly one of the most well known methods for reaction-diffusion type equations since it has fourth-order approach and it has been widely used in practical computational methods. Recently much fitted dinite difference scheme has been studied based on Numerov’s method in [11,12,13,14,15,16,17,18]. In [11], Phaneendra et al gave a finite difference Numerov scheme with a fitted multiplier three bands for solving singularly perturbed boundary value problem. Based on Numerov method, singularly perturbed nonlinear reaction-diffusion problem were investigated in [16,17,18,19].

In this study, we present finite difference scheme based on Numerov method for (1)(2) problem on an uniform mesh. The some properties of the exact solution is given in section 2. In section 3, a finite difference scheme on a uniform mesh is introduced which is based on Numerov’s method. In section 4, the convergence of the approximate solution was presented and it was shown that uniform convergence was achieved at the discrete maximum norm. A numerical example and its results are given in section 5.

Notation

The C symbol in the throughout the article indicates a positive constants and does not have to be the same in each occurence which is independent of ɛ and of the mesh.

2 Properties of the exact solution

The semilinear equation (1) can be written in the take form;

Lu(x)εu(x)+a(x)u(x)=F(x),0<x<l,
u(0)=A,u(l)=B,
where a(x)=f(x,u˜)u , ũ = γu, 0 < γ < 1, F (x) = − f (x, 0).

Here we will give some important properties of the solution of (3)(4) problem, which are required in later sections for the analysis of the numerical solution. We will indicate the maximum norm of any continuous g(x) function on the interval with ‖g.

Definition 1

(Maximum Principle). Assume that v(0) ≥ 0 and v(l) ≥ 0. Then Lv(x) 0, 0 < x < l, implies that v(x) ≥ 0, for all 0 ≤ xl.

The following two lemma and its solutions are given in [22].

Lemma 1

For any v(x) function, let v(x) ∈ C [0, l] ∩ C2 (0, l). Then the following estimate is true.

|v(x)||v(0)|+|v(l)|+α1max1iN|Lv(x)|,0xl.

Proof

Let us define the Ψ(x) function as follows:

Ψ(x)=±v(x)+|v(0)|+|v(l)|+α1max1iN|Lv(x)|,0xl.

Then the following inequalities are satisfied

Ψ(0)0,Ψ(l)0andLΨ(x)0.

The maximum principle gives Ψ(x) ≥ 0, for all 0 ≤ xl, and so the inequalitiy (5) holds.

Lemma 2

Let a(x), F (x) are given sufficiently smooth functions and u(x) be the solution of the problem (3)(4). Then the following estimates hold.

|u(x)||C,0xl.
|u(x)|C{1+1ε(eαxε+eα(lx)ε)}.

Proof

Appliying Lemma 1. to (3)(4) we have (8).

Lv(x)=φ(x),
v(0)=O(1ε),v(l)=O(1ε),
where
v(x)=u(x),φ(x)=F(x)a(x)u(x).

The solution of the problem (10)(11) has the following form:

v(x)=v0(x)+v1(x).

Respectively, in here, the functions v0 (x) and v1 (x) are the solutions of the following problems:

{Lv0(x)=φ(x),0<x<l,v0(0)=v0(l)=0,
{Lv1(x)=0,0<x<l,v1(0)=v1(l)=0,
from Lemma 1., for the solution of the problem (14), we have
|v0(x)|α1max1sl|φ(s)|.

Thus, we obtain

|v0(x)|C,0xl.

Applying maximum principle to the problem (15), we get

|v1(x)|w(x),
where w(x) is the solution of the following problem:
{εw+αw=0,0<x<l,w(0)=|v1(0)|,w(l)=|v1(l)|.

The solution of this problem is has the form

w(x)=1sinh(αlε){|v1(0)|sinh(α(lx)ε)+|v1(l)|sinh(αxε)},
and it is from that
w(x)Cε(eαxε+eα(lx)ε),
is hold. Then combining (16),(17) and (20) in the following inequality, it can be easely obtained:
|u(x)||v0(x)|+|v1(x)|.

Thus the proof is completed.

3 Discretization and Mesh

In this section, we construct a numerical scheme for solving the problem (1)(2) on a uniform mesh. Let wh denote the uniform mesh on [0, l].

wh={xi=ih,i=1,2,,N1,h=1/N},w¯h=wh{x=0,l}.
Let us show wi = w(xi) for any function w(x), and moreover any approximation of the function u(x) at point xi with yi. We will use the following notations for any mesh function {wi} defined on w¯N :
wx¯,i=wiwi1h,wx,i=wi+1wih,wx¯x,i=wx,iwx¯,ih=wi+12wi+wi1h2,
and
wC(w¯h):=max1iN|wi|.

To find the difference approach that corresponds to (1), let us use the following identity and use the interpolating quadrature formulas in [8] on each intervals (xi−1, xi) and (xi, xi+1),

χi1h1xi1xi+1Lu(x)φi(x)dx=0,1iN1,
then we obtain the following relation:
lwiεθiwx¯x,i+f(xi,wi)=Ri,i=1,2,,N1,
where
Ri=χi1h1{xi1xi+1(f(x,w)f(xi,wi)φi(x)dx}.

In here θi is called fitting factor and after a simple calculation, the value of θi=χi1=1 .

if Ri omited in equation (22) then we have numerical scheme for (1)(2)

{lwiεwx¯x,i+f(xi,wi)=0,i=1,2,,N1,w0=A,wN=B.

We note that φi(1) and φi(2) are basis functions such that they are solutions of the following problems respectively,

εφi(1)=0,φi(1)(xi)=1,φi(1)(xi1)=0,εφi(2)=0,φi(2)(xi)=1,φi(2)(xi+1)=0,
and which are
φi(x)={φi(1)(xxi1)h;xi1<x<xi,φi(2)(xi+1x)h;xi<x<xi+1,0,x(xi1,xi+1),
and also χi=h1xi1xi+1φi(x)dx=1 .

3.1 Description of the Numerov’s method

For convenience, let yi = y(xi), y(n)(xi)=yi(n) at x = xi, for any function y(x), using the Taylor series expansion, we obtain:

yi+12yi+yi1h2=yi+112h2yi″″+O(h4).

If the error term is omited, and instead of yi and yi(4) the corresponding terms are written, then we have the following a Numerov finite difference scheme:

(εh2ai112)yi1(2εh2+10ai12)yi+(εh2ai+112)yi+1=112(Fi1+10Fi+Fi+1).i=1,2,,N1.

Let us

Ai=εh2ai112,Ci=2εh2+10ai12,Bi=εh2ai+112,Gi=112(Fi1+10Fi+Fi+1),
then we obtain the followig equation:
Aiyi1Ciyi+Biyi+1=Gi,i=1,2,,N1.

From [10],

Ai>0,Bi>0,CiAiBi>0,
are valid and the siystem has only one solution. This system can be solved by Thomas algorithm.

3.2 An Algorithm for Numerov Type Scheme

If Numerov method is applied to problem (1)(2) we obtain following equation:

{εwx¯x,i+[1+112h2wx¯x,i]f(xj,wj)=0,j=1,2,,N;w0=A,wN+1=B

With the help of the Newton-Raphson-Kantorovich approach, we’ll get a new Numerov type difference scheme. Instead of f (xj, uj) function, in the equation for (28), let’s write the following equivalent,

f(xj,wj(k))=f(xj,wj(k1))+fw(xj,wj(k1)).(wj(k)wj(k1))
then we obtain the following relation.
εwx¯xwj(k)+[1+112h2wx¯x]{f(xj,wj(k1))+fw(xj,wj(k1)).(wj(k)wj(k1))}=0,j=1,2,,N;k=1,2,

If necessary arrangements are made we have the following difference equation.

Aj(k1)wj1(k)Cj(k1)wj(k)+Bj(k1)wj+1(k)=Gj(k1),j=1,2,,N1;k=1,2,
where
Aj(k1)=(εh2112fwj(k1)),Cj(k1)=[2εh2+112(fwj+1(k1)+8fwj(k1)+fwj1(k1))],Bj(k1)=(εh2112fwj(k1)),Gj(k1)=112(fwj+1(k1)+10fwj(k1)+fwj1(k1)).wj(k1)(fj+1(k1)+10fj(k1)+fj1(k1))+fwj(k1).(wj+1(k1)2wj(k1)+wj1(k1))
and
wj+1(0),wj(0),wj1(0)
are given. This scheme has (27) ptoperties and therefore it is stable and has a one solution.

4 Convergence Analysis

Let zi = yiui, 0 ≤ iN, where yi the solution of (22) and ui the solution of (1)(2) at mesh point xi. We now estimate the approximate error zi, which satisfies the following discrete problem

lzi=Ri;z0=zN=0
where the truncation error Ri is in the equation (23).

Definition 2

(Discrete Maximum Principle). Suppose that a mesh function vi satisfies v0 ≥ 0 and vN ≥ 0. Then ℓvi ≥ 0, for all 0 ≤ iN − 1 implies that vi ≥ 0 for all 0 ≤ iN.

Theorem 3

Let f (x, u) ∈ C2 [0, l]. Then the following estimate holds.

yuC(w¯h)Ch2

Proof

If we find an estimate for Ri the proof is complete. For function f (x, u), using the Taylor series expansion, we obtain:

f(x,u)f(xi,ui)=(xxi){f(xi,ui)x+f(xi,ui)udu(xi)dx}++(xxi)22!{2f(ξi,u(ξi))x2+2f(ξi,u(ξi))xudu(ξi)dx+2f(ξi,u(ξi))u2(du(ξi)dx)2}

If we write this expression instead of Ri we obtain the following relation:

(xxi){f(xi,ui)x+f(xi,ui)udu(xi)dx}++(xxi)22!(2f(ξi,u(ξi))x2+2f(ξi,u(ξi))xudu(ξi)dx+2f(ξi,u(ξi))u2(du(ξi)dx)2)φi(x)dx

Considering the equivalents of u and u, using of discrete maximum principle we have

RC(w¯h)Ch2

This estimate conclude the proof of Theorem 4.

5 Numerical example

To illustrate the applicability of the method proposed in this article, we applied it to an example. Connsider the following semilinear problem:

Example 4.ɛuexp(−(x2 + u)) = 0, x ∈ [0, 1], u(0) = 0, u(1) = 1

For numerical approximation of solution, we have shown that the method uniform convergens according to the ɛ-parameter. Since the exact solution to this problem could not be found, we used the following the double mesh prensiple for calculate of the maximum absolute errors.

EεN=max0iN|uiNu2i2N|

For any N, the ɛ-uniform maximum absolute error is calculated by

EN=maxεEε.

The numerical rate of convergence and ɛ-uniform convergence rate for example has been calculated by the following formulas;

RεN=log|EεN/Eε2N|log2,RN=log|EN/E2N|log2

The maximum point wise errors and the rates of convergence of the problem in example is presented in Table 1. ( ui0=x2 , 1 ≤ iN − 1, for arbitrary inital function)

Table 1

Maximum errors and rates of convergence wh

ɛN = 32N = 64N = 128N = 256N = 512RεN
2−52.3e–45.9e–51.5e–53.7e–69.2e–72.009
2−63.8e–49.5e–52.4e–56.2e–61.5e–62.014
2−76.4e–41.6e–44.0e–51.0e–52.5e–62.019
2−81.1e–32.4e–46.9e–51.7e–54.3e–62.020
2−91.9e–34.9e–41.2e–43.1e–57.6e–62.015
RN2.0132.0032.0021.9982.020

6 Conclusion

In this paper we have presented a Numerov’s scheme to solve a class of singularly perturbed semilinear reaction-diffusion problem. We have introduced computational technique based Numerov’s scheme on a uniform mesh. Uniform convergence of the method is demonstrated with respect to the parameter of perturbation. The accuracy of the uniform convergence was supported by a numerical example.

Acknowledgements

This work was supported by Research Fund of the Yuzuncu Yil University. Project Number: FDK-2017-5843.

References

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    R. E. O'Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.

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    E. R. Doolan, J.J.H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, 1980.

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    H.-G. Roos, Layer-adapted grids for singular perturbation problems, ZAMM Z Angew Math Mech 78 (1998), 291–309.

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    J. H. Miller, E. O'Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.

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    G.M. Amiraliyev, Y.D, Mamedov, Differences Schemes on the Uniform Mesh for Singular Perturbed Pseudo-Parabolic Equations, Tr. J. of Mathematics, 19, 3 (1995), 207–222.

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    G.M. Amiraliyev, The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system, Applied Mathematics and Computation, 162, 3 (2005) 1023–1034.

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    G.M. Amiraliyev, F. Erdogan, Difference schemes for a class of singularly perturbed initial value problems for delay differential equations, Numer. Algorithms, 52, 4 (2009) 663–675.

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    Samarskii A.A., Theory of difference schemes. Monographs and textbooks in pure and applied mathematics v 240. Marcel Dekker, New York, 761., 2001.

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    K. Phaneendra, P. Pramod Chakravarthy, Y. N. Reddy, A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers, Applied Mathematics and Information Sciences 4, 3, (2010) 341–352.

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    Holevoet, D., Daele, M.V., Berghe, G.V., The Optimal Exponentially-Fitted Numerov Method for Solving Two-Point Boundary Value Problems. Journal of Comp. And Applied Mathematics, 230, (2010) 260–269.

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    K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems, Applied Math. and Comp. 171 (2005) 547–566.

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    R.K. Bawa, A Paralel aproach for self-adjoint singular perturbation problems using Numerov’s scheme, nternational Journal of Computer Math. Vol. 84, No. 3 (2007) 317–323.

  • [15]

    Linss, T., Sufficient Conditions for Uniform Convergence on LAyer-Adapted Meshes for One-Dimentional Reaction-Diffusion Problems. Numerical Algorithms, 40 :(2005) 23–32.

  • [16]

    Wang, Y.M., On Numerov’s Method for a Class of Strongly Nonlinear Two-Point Boundary Value Problems, Applied Numerical Mathematics, 61, (2011) 38–52.

  • [17]

    Stynes, M., Kopteva, N., Numerical Analysis of Singularly Perturbed Nonlinear Reaction-Diffusion Problems with Multiple Solutions. Computers and Mathematics with Applications, 51 : (2006) 857–864.

  • [18]

    Wang, Y.M., On Numerov’s Method for a Class of Strongly Nonlinear Two-Point Boundary Value Problems. Applied Numerical Mathematics, 61 :(2011) 38–52.

  • [19]

    Wang, Y.M., Wu, W.J., and Scalia, M., Numerov’s Method for a Class of Nonlinear Multipoint Boundary Value Problems. Hundawi Publishing Corporation, Mathematical Problems in Engineering, (2012) : Article ID 316852, 29pp.

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    Kopteva, N., Stynes, M., Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions. Applied Numerical Mathematics, 51: 2–3, (2004) 273–288.

  • [21]

    I.G. Amiraliyeva, F. Erdogan, G.M. Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay inital value problem. Applied Mathematics Letters, 23: (2010) 1221–1225. 0z0

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    G.M. Amirali, I. Amirali, Numerical Analysis(In Turkish); I. Edition, Seckin Publications, Ankara, 2018.

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  • [1]

    R. E. O'Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.

  • [2]

    E. R. Doolan, J.J.H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, 1980.

  • [3]

    P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers; Chapman-Hall/CRC, New York, 2000.

  • [4]

    H.-G. Roos, Layer-adapted grids for singular perturbation problems, ZAMM Z Angew Math Mech 78 (1998), 291–309.

  • [5]

    H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, 1996.

  • [6]

    J. H. Miller, E. O'Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.

  • [7]

    G.M. Amiraliyev, Y.D, Mamedov, Differences Schemes on the Uniform Mesh for Singular Perturbed Pseudo-Parabolic Equations, Tr. J. of Mathematics, 19, 3 (1995), 207–222.

  • [8]

    G.M. Amiraliyev, The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system, Applied Mathematics and Computation, 162, 3 (2005) 1023–1034.

  • [9]

    G.M. Amiraliyev, F. Erdogan, Difference schemes for a class of singularly perturbed initial value problems for delay differential equations, Numer. Algorithms, 52, 4 (2009) 663–675.

  • [10]

    Samarskii A.A., Theory of difference schemes. Monographs and textbooks in pure and applied mathematics v 240. Marcel Dekker, New York, 761., 2001.

  • [11]

    K. Phaneendra, P. Pramod Chakravarthy, Y. N. Reddy, A Fitted Numerov Method for Singular Perturbation Problems Exhibiting Twin Layers, Applied Mathematics and Information Sciences 4, 3, (2010) 341–352.

  • [12]

    Holevoet, D., Daele, M.V., Berghe, G.V., The Optimal Exponentially-Fitted Numerov Method for Solving Two-Point Boundary Value Problems. Journal of Comp. And Applied Mathematics, 230, (2010) 260–269.

  • [13]

    K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems, Applied Math. and Comp. 171 (2005) 547–566.

  • [14]

    R.K. Bawa, A Paralel aproach for self-adjoint singular perturbation problems using Numerov’s scheme, nternational Journal of Computer Math. Vol. 84, No. 3 (2007) 317–323.

  • [15]

    Linss, T., Sufficient Conditions for Uniform Convergence on LAyer-Adapted Meshes for One-Dimentional Reaction-Diffusion Problems. Numerical Algorithms, 40 :(2005) 23–32.

  • [16]

    Wang, Y.M., On Numerov’s Method for a Class of Strongly Nonlinear Two-Point Boundary Value Problems, Applied Numerical Mathematics, 61, (2011) 38–52.

  • [17]

    Stynes, M., Kopteva, N., Numerical Analysis of Singularly Perturbed Nonlinear Reaction-Diffusion Problems with Multiple Solutions. Computers and Mathematics with Applications, 51 : (2006) 857–864.

  • [18]

    Wang, Y.M., On Numerov’s Method for a Class of Strongly Nonlinear Two-Point Boundary Value Problems. Applied Numerical Mathematics, 61 :(2011) 38–52.

  • [19]

    Wang, Y.M., Wu, W.J., and Scalia, M., Numerov’s Method for a Class of Nonlinear Multipoint Boundary Value Problems. Hundawi Publishing Corporation, Mathematical Problems in Engineering, (2012) : Article ID 316852, 29pp.

  • [20]

    Kopteva, N., Stynes, M., Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions. Applied Numerical Mathematics, 51: 2–3, (2004) 273–288.

  • [21]

    I.G. Amiraliyeva, F. Erdogan, G.M. Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay inital value problem. Applied Mathematics Letters, 23: (2010) 1221–1225. 0z0

  • [22]

    G.M. Amirali, I. Amirali, Numerical Analysis(In Turkish); I. Edition, Seckin Publications, Ankara, 2018.

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