Numerical Simulation for Singularly Perturbed Problem with Two Nonlocal Boundary Conditions

Year 2022, Volume: 14 Issue: 2, 235 - 247, 30.12.2022

Derya Arslan , Musa Çakır

https://doi.org/10.47000/tjmcs.862848

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

• 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.
• Arslan, D., An approximate solution of linear singularly perturbed problem with nonlocal boundary condition, Journal of Mathematical Analysis, 11(2020), 46–58.
• 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.
• Benchohra, M., Ntouyas, S.K., Existence of solutions of nonlinear differential equations with nonlocal conditions, J. Math. Anal. Appl., 252(2000), 477–483.
• Bender, C.M., Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
• Bougoffa, L., Khanfer, A., Existence and uniqueness theorems of second-order equations with integral boundary conditions, 55(2018), 899-911.
• 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).
• Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, 162(1991), 494–501.
• Cakir, M., Arslan, D., A new numerical approach for a singularly perturbed problem with two integral boundary conditions, Computational and Applied Mathematics, 40(2021).
• 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.
• 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.
• Chegis, R., The numerical solution of singularly perturbed nonlocal problem (in Russian), Lietuvas Matematica Rink, 28(1988), 144–152.
• Doolan, E.P., Miller, J.J.H., Schilders,W.H.A., Uniform Numerical Method for Problems with Initial and Boundary Layers, Boole Press, 1980.
• 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.
• 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.
• 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.
• Kevorkian, J., Cole, J.D., Perturbation Methods in Applied Mathematics, Springer, New York, 1981.
• 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).
• 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.
• Linss, T., Stynes, M., A hybrid difference on a Shishkin mesh linear convection-diffusion problems, Applied Numerical Mathematics, 31(1999), 255–27.
• Linss, T., Layer-adapted meshes for convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 192(2003), 1061–1105.
• Miller, J.J.H., O’Riordan, E, Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
• Nayfeh, A.H., Perturbation Methods, Wiley, New York, 1985.
• Nayfeh, A.H., Problems in Perturbation, Wiley, New York, 1979.
• O’Malley, R.E., Singular Perturbation Methods for Ordinary Differential Equations, Springer Verlag, New York, 1991.
• 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.
• Roos, H.G., Stynes, M., Tobiska, L., Robust Numerical Methods Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, 2008.
• Samarskii, A.A., Theory of Difference Schemes, 2nd ed., ”Nauka”, Moscow, 1983.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.