Back
 JAMP  Vol.4 No.6 , June 2016
A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem
Abstract: We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter ε). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.
Cite this paper: Çakır, M. and Arslan, D. (2016) A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem. Journal of Applied Mathematics and Physics, 4, 1143-1156. doi: 10.4236/jamp.2016.46119.
References

[1]   Cziegis, R. (1988) The Nümerical Solution of Singularly Pertürbed Nonlocal Problem. Lietuvos Matematikos Rinkinys, 28, 144-152. (In Russian)

[2]   Cziegis, R. (1991) The Difference Schemes for Problems with Nonlocal Conditions. Informatica (Lietuva), 2, 155-170.

[3]   Bakhvalov, N.S. (1969) On Optimization of Methods for Solving Boundary-Value Problems in the Presence of a Boundary Layer. The Use of Special Transformations in the Numerical Solution of Bounary-Layer Problems. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 9, 841-859.

[4]   Amiraliyev, G.M. and Cakir, M. (2007) Numerical Solution of a Singularly Perturbed Three-Point Boundary Value Problem. International Journal of Applied Mathematics, 84, 1465-1481.

[5]   Amiraliyev, G.M. and Çakir, M. (2000) A Uniformily Convergent Difference Scheme for Singularly Perturbed Problem with Convective Term End Zeroth Order Reduced Equation. International Journal of Applied Mathematics, 2, 1407-1419.

[6]   Amiraliyev, G. M. and Çakir, M. (2002) Numerical Solution of the Singularly Perturbed Problem with Nonlocal Boundary Condition. Applied Mathematics and Mechanics (English Edition), 23, 755-764.
http://dx.doi.org/10.1007/BF02456971

[7]   Amiraliyev, G.M. and Duru, H. (2005) A Note on a Parameterized Singular Perturbation Problem. Journal of Computational and Applied Mathematics, 182, 233-242.
http://dx.doi.org/10.1016/j.cam.2004.11.047

[8]   Amiraliyev, G.M. and Erdogan, F. (2007) Uniform Numerical Method for Singularly Perturbed Delay Differential Equations. Computers & Mathematics with Applications, 53, 1251-1259.
http://dx.doi.org/10.1016/j.camwa.2006.07.009

[9]   Amiraliyeva, I.G., Erdogan, F. and Amiraliyev, G.M. (2010) A Uniform Numerical Method for Dealing with a Singularly Perturbed Delay Initial Value Problem. Applied Mathematics Letters, 23, 1221-1225.
http://dx.doi.org/10.1016/j.aml.2010.06.002

[10]   Adzic, N. and Ovcin, Z. (2001) Nonlinear Spp with Nonlocal Boundary Conditions and Spectral Approximation. Novi Sad Journal of Mathematics, 31, 85-91.

[11]   Amiraliyev, G.M., Amiraliyeva, I.G. and Kudu, M. (2007) A Numerical Treatment for Singularly Perturbed Differential Equations with Integral Boundary Condition. Applied Mathematics and Computations, 185, 574-582.
http://dx.doi.org/10.1016/j.amc.2006.07.060

[12]   Herceg, D. (1990) On the Numerical Solution of a Singularly Perturbed Nonlocal Problem. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 20, 1-10.

[13]   Herceg, D. (1991) Solving a Nonlocal Singularly Perturbed Problem by Splines in Tension. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat, 21, 119-132.

[14]   Çakir, M. (2010) Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem. Hindawi Publising Corporation Advances in Difference Equations, Vol. 2010, 13 p.

[15]   Geng, F.Z. (2012) A Numerical Algorithm for Nonlinear Multi-Point Boundary Value Problems. Journal of Computational and Applied Mathematics, 236, 1789-1794.
http://dx.doi.org/10.1016/j.cam.2011.10.010

 
 
Top