Numerical Simulation for Singularly Perturbed Problem with Two Nonlocal Boundary Conditions

Abstract

In this paper, numerical solution for singularly perturbed problem with nonlocal boundary conditions is obtained. Finite difference method is used to discretize this problem on the Bakhvalov-Shishkin mesh. The some properties of exact solution are analyzed. The error is obtained first-order in the discrete maximum norm. Finally, an example is solved to show the advantages of the finite difference method.

Keywords

• Singular perturbation equation
• finite difference method
• Bakhvalov-Shishkin mesh
• uniform convergence
• integral conditions

References

1. Amiraliyev, G.M., Mamedov, Y.D., Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turkish Journal of Mathematics, 19(1995), 207–222.
2. Arslan, D., An approximate solution of linear singularly perturbed problem with nonlocal boundary condition, Journal of Mathematical Analysis, 11(2020), 46–58.
3. Arslan, D., An effective numerical method for singularly perturbed nonlocal boundary value problem on Bakhvalov Mesh, Journal of Informatics and Mathematical Sciences, 11(2019), 253–264, 2019.
4. Benchohra, M., Ntouyas, S.K., Existence of solutions of nonlinear differential equations with nonlocal conditions, J. Math. Anal. Appl., 252(2000), 477–483.
5. Bender, C.M., Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
6. Bougoffa, L., Khanfer, A., Existence and uniqueness theorems of second-order equations with integral boundary conditions, 55(2018), 899-911.
7. Bulut, H., Akturk, T., Ucar, Y., The solution of advection diffusion equation by the finite elements method, International Journal of Basic and Applied Sciences IJBAS-IJENS, 13(2013).
8. Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, 162(1991), 494–501.

9. Cakir, M., Arslan, D., A new numerical approach for a singularly perturbed problem with two integral boundary conditions, Computational and Applied Mathematics, 40(2021).
10. Cakir, M., A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition, Mathematical Modelling and Analysis, 21(2016), 644–658.
11. Cakir, M., Amiraliyev, G.M., Numerical solution of the singularly perturbed three-point boundary value problem, International Journal of Computer Mathematics, 84(2007), 1465–1481.
12. Chegis, R., The numerical solution of singularly perturbed nonlocal problem (in Russian), Lietuvas Matematica Rink, 28(1988), 144–152.
13. Doolan, E.P., Miller, J.J.H., Schilders,W.H.A., Uniform Numerical Method for Problems with Initial and Boundary Layers, Boole Press, 1980.
14. Farell, 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.
15. Ilhan, O.A., Kasimov, S.G., Madraximov, U.S., Baskonus, H.M., On solvability of the mixed problem for a partial differential equation of a fractional order with Sturm-Liouville operators and non-local boundary conditions, Rocky Mountain Journal of Mathematics, 49(2019), 1191–1206.
16. Kaya, D., Bulut, H., On a numerical comparison of decomposition method and finite difference method for an elliptic P.D.E, F.Ü. Fen ve MÜh.Bil. Dergisi, 11(1999), 285–294.
17. Kevorkian, J., Cole, J.D., Perturbation Methods in Applied Mathematics, Springer, New York, 1981.
18. Khan, R.A., The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 10(2003).
19. Kumar, D., Kumari, P., A parameter-uniform collocation scheme for singularly perturbed delay problems with integral boundary condition, Journal of Applied Mathematics and Computing, 63(2020), 813–828.
20. Linss, T., Stynes, M., A hybrid difference on a Shishkin mesh linear convection-diffusion problems, Applied Numerical Mathematics, 31(1999), 255–27.
21. Linss, T., Layer-adapted meshes for convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 192(2003), 1061–1105.
22. Miller, J.J.H., O’Riordan, E, Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
23. Nayfeh, A.H., Perturbation Methods, Wiley, New York, 1985.
24. Nayfeh, A.H., Problems in Perturbation, Wiley, New York, 1979.
25. O’Malley, R.E., Singular Perturbation Methods for Ordinary Differential Equations, Springer Verlag, New York, 1991.
26. Raja, V., Tamilselvan, A., Fitted finite difference method for third order singularly perturbed convection diffusion equations with integral boundary condition, Arab Journal of Mathematical Science, 25(2019), 231–242.
27. Roos, H.G., Stynes, M., Tobiska, L., Robust Numerical Methods Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, 2008.
28. Samarskii, A.A., Theory of Difference Schemes, 2nd ed., ”Nauka”, Moscow, 1983.
29. Savin, I.A., On the rate of convergence, uniform with respect to a small parameter of a difference scheme for an ordinary differential equation, Computational Mathematics and Mathematical Physics, 35(1995), 1417–1422.
30. Sekar, E., Tamilselvan, A., Singularly perturbed delay differential equations of convection-diffusion type with integral boundary condition, Journal of Applied Mathematics and Computing, 59(2019), 701–722.
31. Sekar, E., Tamilselvan, A., Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition, Journal of Applied Mathematics and Computational Mechanics, 18(2019), 99–110.
32. Xie, F,. Jin, Z., Ni, M., On the step-type contrast structure of a second-order semilinear differential equation with integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 62(2010), 1–14.
33. Zheng, Q., Li, X., Liu, Y., Uniform second-order hybrid schemes on Bakhvalov-Shishkin mesh for quasi-linear convection-diffusion problems, Advanced Materials Research, 871(2014), 135–140.
34. Zhou, P., Yin, Y., Yang,Y., Finite element super convergence on Bakhvalov- Shishkin mesh for singularly perturbed problem, Journal on Numerical Methods and Computer Applications, 34(2013), 257–265.