Local Annihilation Method and Some Stiff Problems
Subject Areas : International Journal of Industrial Mathematics
A. Abdollahi
1
,
E. Babolian
2
1 - Department of Mathematics, Maragheh Branch, Islamic Azad University, Maragheh, Iran.
2 - Department of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.
Keywords: Residual function, Stiff problems, Ill-posed problems, Local annihilation method,
Abstract :
In this article, a new scheme inspired from collocation method is presented for numerical solution of stiff initial-value problems and Fredholm integral equations of the first kind based on the derivatives of residual function. Then, the error analysis of this method is investigated by presenting an error bound. Numerical comparisons indicate that the presented method yields accurate approximations in many cases in which the collocation method is failed.
[1] A. Abdollahi, E. Babolian, Theory of blockpulse functions in numerical solution of Fredholm integral equations of the second kind,Int. J. Industrial Mathematics 8 (2016) 157-163.
[2] R. R. Ahmad, N. Yaacob, Third-order composite Runge-Kutta method for stiff problems, International Journal of Computer Mathematics 82 (2006) 1221-1226.
[3] E. Babolian, A. Abdollahi, S. Shahmorad, Chain least squares method and ill-posed problems, Iranian Journal of Science & Technology 38 (2014) 123-132.
[4] E. Babolian, L. M. Delves, An augmented Galerkin method for first kind Fredholm equations, J. Inst. Maths. Applics 24 (1979) 157-174.
[5] E. Babolian, T. Lotfi, M. Paripour, Wavelet moment method for solving Fredholm integral equations of the first kind, Applied Mathematics and Computation 186 (2007) 1467-1471.
[6] C. A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, (1989).
[7] A. V. Bitsadze, Integral Equations of First Kind, World Scientific Publishing Co. Pte. Ltd., (1995).
[8] N. M. Bujurke, C. S. Salimath, S. C. Shiralashetti, Numerical solution of stiff systems from nonlinear dynamics using singleterm Haar wavelet series, Nonlinear Dynamics 51 (2008) 595-605.
[9] B. N. Datta, Numerical Linear Algebra and Applications Second Edition, SIAM, (2010).
[10] L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, (1985).
[11] H. Ernst, W. Gerhard, Solving ordinary differential equations II: Stiff and differentialalgebraic problems, Springer-Verlag, New York, (1996).
[12] B. Faleichik, I. Bondar, V. Byl, Generalized Picard iterations: A class of iterated Runge Kutta methods for stiff problems, Journal of Computational and Applied Mathematics 262 (2014) 37-50.
[13] P. A. Farrell, Uniform and optimal schemes for stiff initial value problems, Comput. Math. Applic. 13 (1987) 925-936.
[14] S. Faure, M. M. Tekitek, R. Temam, Finite volume approximation of stiff problems on 30 A. Abdollahi et al., /IJIM Vol. 12, No. 1 (2020) 23-30 two-dimensional curvilinear domain, International Journal of Computer Mathematics 93 (2015) 1787-1799.
[15] F. Goharee, E. Babolian, A. Abdollahi, Modified chain least squares method and some numerical results, Iranian Journal of Science & Technology 38 (2014) 91-99.
[16] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics, Boston MA, (1984).
[17] C. Hsiao, Numerical solution of stiff differential equations via Haar wavelets, International Journal of Computer Mathematics 82 (2006) 1117-1123.
[18] R. Kress, Linear Integral Equations, Springer-Verlag, (1999).
[19] R. Kress, Numerical Analysis, SpringerVerlag, New York, (1998).
[20] H. N. Mhaskar, D. V. Pai, Fundamentals of Approximation Theory, DAlpha Science International Ltd. (2000).
[21] P. Novati, A class of explicit one-step methods of order two for stiff problems, Journal of Numerical Mathematics 13 (2005) 219-236.
[22] R. I. Okuonghae, M. N. O. Ikhile, A class of hybrid linear multistep methods with Astability properties for stiff IVPs in ODEs, Journal of Numerical Mathematics 21 (2013) 157-172.
[23] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Ass. Comput. Mach. 9 (1962) 84-96.
[24] J. I. Ramos, Linearization techniques for singularity-perturbed initial-value problems of ordinary differential equations, Applied Mathematics and Computation 163 (2005) 1143-1163.
[25] W. Rudin, Principles of Mathematical Analysis 3nd ed., McGraw-Hill, New York, (1976).
[26] R. E. Scraton, Some L-stable methods for stiff differential equations, International Journal of Computer Mathematics 9 (2007) 81-87.
[27] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems Dordrecht, Boston, (1995).