An Innovative Technique for Solving Singularly Perturbed Problems with Integral Boundary Conditions on the Non-Uniform Mesh

Abstract

We investigate an innovative numerical method for solving linear singularly perturbed problems with integral boundary conditions. Firstly, proving a uniformly convergent numerical method for solving the singularly perturbed problem is the main goal of this problem. Secondly, a piecewise equidistant mesh is used to generate the finite difference scheme. Next, the difference technique's stability and convergence analysis are covered. Lastly, two test instances' numerical results are shown.

Keywords

• Linear Singularly Perturbed Problem
• Non-Uniform Mesh
• Numerical Method
• Integral Boundary.

References

1. [1] Adzic, N.: Spectral approximation and nonlocal boundary value problems. Novi Sad J. Math. 30, 1-10 (2000)
2. [2] Amiraliyev, G.M., Mamedov, Y.D.: Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations. Turkish J. Math., 19, 207-222 (1995)
3. [3] Amiraliyev, G.M., Amiraliyeva I.G, and Kudu M.: A Numerical Treatment for Singularly Perturbed Differential Equations with Integral Boundary Condition. Appl. Math. Comput., 185, 574-582 (2007)
4. [4] Bakhvalov, N.S.: The optimization of methods of solving boundary value problems with a boundary layer. USSR Comp. Math. Math. Phys. 9, 139-166 (1969)
5. [5] Benchohra, M., Ntouyas, S.K.: Existence of solutions of nonlinear differential equations with nonlocal conditions. J. Math. Analy. Appl. 252(1), 477-483 (2000)
6. [6] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Analy. Applicat. 162(2), 494-505 (1991)
7. [7] Cakir, M.: Uniform second-order difference method for a singularly perturbed three-point boundary value problem. Advances in Diff Equ. 2010, 1-13 (2010)
8. [8] Cakir, M., Amiraliyev, G.M.: A second order numerical method for singularly perturbed problem with non-local boundary condition. Journal of Applied Mathematics and Computing, 67(1), 919-936 (2021)

9. [9] Chegis, R.: The Numerical solution of singularly perturbed nonlocal problem. Lietuvos Matematikos Rinkinys (in Russian), 28, 144-152 (1988)
10. [10] Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin (1980)
11. [11] Farrel, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman Hall/CRC, New York (2000)
12. [12] Gupta, C.P., Trofimchuk, S.I.: A sharper condition for the solvability of a three-point second order boundary value problem. J. Math. Anal and Appl. 205, 586-597 (1997)
13. [13] Herceg, D.: On the numerical solutions of a singularly perturbed nonlocal problem. Univ. U Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Math., 20, 1-10 (1990)
14. [14] Jankowski T. Extensions of quasilinearization method for differential equations with integral boundary conditions. Math Comput Modell, 37, 155-165 (2003)
15. [15] Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics. Springer, New York (1981)
16. [16] Kopteva, N., and O’Riordan, E.: Shishkin meshes in the numerical solution of singularly perturbed differential equations. Int. J. Num. Analy. Mod., 7, 393-415 (2010)
17. [17] Kudu M, Amirali I.G, Amiraliyev, G.M.: A Layer Analysis of Parameterized Singularly Perturbed Boundary Value Problem. IJAM, 29(4), 439-449 (2016)
18. [18] Kudu M, Amirali I.G, Amiraliyev, G.M.: Uniform numerical approximation for parameter dependent singularly perturbed problem with integral boundary condition. Miskolc Mathematical Notes, 19(1), 337-353 (2018)
19. [19] Linss, T.: Layer-adapted meshes for convection-diffusion problems. Comput Meth. Appl. Mech. and Eng., 192(9-10), 1061-1105 (2003)
20. [20] Linss, T., Stynes, M.: A hybrid difference on a Shishkin mesh linear convection-diffusion problems. Applied Numer. Math., 31(3), 255-270 (1999).
21. [21] Miller, JJH., O’Riordan, E., and Shishkin, GI., 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)
22. [22] Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1993)
23. [23] O’Malley, R.E.: Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York (1991)
24. [24] Roos, HG., Stynes M., Tobiska L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. 2nd ed. Springer Verlag, Berlin (2008)
25. [25] Sapagovas, M., Chegis R.: On some boundary value problems with nonlocal condition. in Russian, Differ. Equ. 23, 1268-1274 (1987)
26. [26] Shishkin, G.I.: Grid approximation of a singularly perturbed elliptic convection-diffusion equation in an unbounded domain, Russ. J. Numer. Aanal. Math. Modell., 21 , 67-94 (2006)
27. [27] Smith, D.R.: Singular Perturbation Theory. Cambridge University Press, Cambridge (1985)
28. [28] Stynes, M., Roos, H.G., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer Verlag, Berlin (2008)
29. [29] Temel, Z. and Cakir, M..: A Robust Numerical Method for a Singularly Perturbed Semilinear Problem with Integral Boundary Conditions. Contemp. Math., 5 , 446-464 (2024)
30. [30] Temel, Z. and Cakir, M.: A New Numerical Scheme for Singularly Perturbed Reaction Diffusion Problems. Gazi Uni. J. Sci., 36, 792-805 (2023)