An approximate solution of singularly perturbed problem on uniform mesh
In this study, we obtain approximate solution for singularly perturbed problem of differential equation having two integral boundary conditions. With this purpose, we propose a new finite difference scheme. First, we construct this exponentially difference scheme on a uniform mesh using the finite difference method. We use the quasilinearization method and the interpolating quadrature formulas to establish the numerical scheme. Then, as a result of the error analysis, we show that the method under study is convergent in the first order. Consequently, theoretical findings are supported by numerical results obtained with an example. Approximate solutions curves are compared on the chart to provide concrete indication. The maximum errors and convergence rates obtained are given on the table for different varepsilon and N values.
[1] Cakir, M., Amiraliyev, G.M. (2005) A finite differ- ence method for the singularly perturbed problem with nonlocal boundary condition. Applied Mathemat- ics and Computation, 160, 539-549.
[2] Amiraliyev, G.M., Cakir, M. (2000). A uniformily convergent difference scheme for singularly perturbed problem with convective term and zeroth order re- duced equation. International Journal of Applied Mathematics, 2(12), 1407-1419.
[3] Cakir, M. (2010). Uniform second-order difference method for a singularly perturbed three-point bound- ary value problem. Advances in Difference Equations,13 pages.
[4] Cakir, M., Amiraliyev, G.M. (2010). A numerical method for a singularly perturbed three-point bound- ary value problem. Journal of Applied Mathematics,17 pages.
[5] Amiraliyev, G.M., Mamedov, Y.D. (1995). Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations. Turkish Journal of Math- ematics, 19, 207-222.
[6] Arslan, D., Cakir, M. (2021). A new numerical ap- proach for a singularly perturbed problem with two integral boundary conditions. Computational and Ap- plied ,mathematics, 40(6).
[7] Arslan, D. (2020). An approximate solution of linear singularly perturbed problem with nonlocal boundary condition. Journal of Mathematical Analysis, 11(3), 46-58.
[8] Arslan, D. (2019). An effective numerical method for singularly perturbed nonlocal boundary value prob- lem on Bakhvalov Mesh. Journal of Informatics and Mathematical Sciences, 11(3-4), 253-264.
[9] Arslan, D. (2019). A novel hybrid method for singu- larly perturbed delay differential equations. Gazi Uni- versity Journal of Science, 32(1), 217-223.
[10] Arslan, D. (2019). Approximate solutions of singu- larly perturbed nonlinear Ill-posed and sixth-order Boussinesq equations with hybrid method. Bitlis Eren U(¨)niversitesi Fen Bilimleri Dergisi, 8(2), 451-458.
[11] Negero, N.T, Duressa, G.F. (2021). A method of line with improved accuracy for singularly perturbed par- abolic convection–diffusion problems with large tem- poral lag. Results in Applied Mathematics, 11, 100174.
[12] Negero, N.T. (2022). A uniformly convergent numer- ical scheme for two parameters singularly perturbed parabolic convection–diffusion problems with a large temporal lag. Results in Applied Mathematics, 16, 100338.
[13] Negero, N.T. (2023). A parameter-uniform efficient numerical scheme for singularly perturbed time-delay parabolic problems with two small parameters. Par- tial Differential Equations in Applied Mathematics, 7, 100518.
[14] Negero, N.T. (2023). A robust fitted numerical scheme for singularly perturbed parabolic reaction–diffusion problems with a general time delay. Results in Physics, 51, 106724.
[15] Negero, N.T. (2023). A fitted operator method of line scheme for solving two-parameter singularly per- turbed parabolic convection-diffusion problems with time delay. Journal of Mathematical Modeling, 11(2).
[16] Nayfeh, A.H. (1985). Perturbation Methods. Wiley, New York.
[17] Nayfeh, A.H. (1979). Problems in Perturbation. Wiley, New York.
[18] Kevorkian, J., Cole, J.D. (1981). Perturbation Meth- ods in Applied Mathematics. Springer, New York.
[19] O’Malley, R.E. (1991). Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York.
[20] Miller, J.J.H., O’Riordan, E., Shishkin, G.I. (1996). Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore.
[21] Roos, H.G., Stynes, M., Tobiska, L. (2008). Robust Numerical Methods Singularly Perturbed Differential Equations. Springer-Verlag, Berlin.
[22] Bakhvalov, N.S. (1969). On optimization of methods for solving boundary value problems in the presence of a boundary layer. The use of special transforma- tion the numerical solution of boudary-layer problems. Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, 9(4) 841-859.
[23] Bitsadze, A.V., Samarskii, A.A. (1969). On Some Sim- pler Generalization of Linear Elliptic Boundary Value Problems. Doklady Akademii Nauk SSSR, 185, 739- 740.
[24] Herceg, D., Surla, K. (1991). Solving a nonlocal singu- larly perturbed nonlocal problem by splines in tension. Univ. u Novom Sadu Zb. Rad.Prirod.-Mat. Fak. Ser. Math., 21(2), 119-132.
[25] Gupta, C.P., Trofimchuk, S.I. (1997). A sharper con- dition for the solvability of a three-point second or- der boundary value problem. Journal of Mathematical Analysis and Applications, 205 586-597.
[26] Chegis, R. (1988). The numerical solution of singu- larly perturbed nonlocal problem (in Russian). Lietu- vas Matematica Rink, 28, 144-152.
[27] Chegis, R. (1991). The difference scheme for prob- lems with nonlocal conditions, Informatica (Lietuva), 2, 155-70.
[28] Nahushev, A.M. (1985). On nonlocal boundary value problems (in Russian). Differential Equations, 21, 92- 101.
[29] Sapagovas, M., Chegis, R. (1987). Numerical solution of nonlocal problems (in Russian). Lietuvas Matemat- ica Rink, 27, 348-356.
[30] Xie, F., Jin, Z., Ni, M. (2010). On the step-type con- trast structure of a second-order semilinear differen- tial equation with integral boundary conditions. Elec- tronic Journal of Qualitative Theory of Differential Equations, 62, 1-14.
[31] Kumar, D., Kumari, P. (2020). A parameter-uniform collocation scheme for singularly perturbed delay problems with integral boundary condition. Journal of Applied Mathematics and Computing, 63, 813-828.
[32] Sekar, E., Tamilselvan, A. (2019). Third order sin- gularly perturbed delay differential equation of reac- tion diffusion type with integral boundary condition. Journal of Applied Mathematics and Computational Mechanics, 18(2), 99-110.
[33] Raja, V., Tamilselvan, A. (2019). Fitted finite dif- ference method for third order singularly perturbed convection diffusion equations with integral bound- ary condition. Arab Journal of Mathematical Science, 25(2), 231-242.
[34] Khan, R.A. (2003). The generalized method of quasi- linearization and nonlinear boundary value problems with integral boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations, 10, 1- 9.
[35] Rao, S.C.S., Kumar, M. (2007). B-spline colloca- tion method for nonlinear singularly perturbed two- point boundary-value problems. Journal of Optimiza- tion Theory and Applications, 134(1), 91-105.
[36] Byszewski, L. (1991). Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Journal of Mathematical Analysis and Applications, 62, 494-505.
[37] Bougoffa, L., Khanfer, A. (2018). Existence and uniqueness theorems of second-order equations with integral boundary conditions. Bulletin of the Korean Mathematical Society, 55(3), 899-911.
[38] Benchohra, M., Ntouyas, S.K. (2000). Existence of so- lutions of nonlinear differential equations with nonlo- cal conditions. Journal of Mathematical Analysis and Applications, 252, 477-483.
[39] Samarskii, A.A. (1983). Theory of Difference Schemes.2 nd ed., ”Nauka”, Moscow.