Rational approximations for solving cauchy problems

Year 2016, Volume: 4 Issue: 3, 254 - 262, 30.09.2016

Veyis Turut Mustafa Bayram

Abstract

In this letter, numerical solutions of Cauchy problems
are considered by multivariate Padé approximations (MPA). Multivariate Padé
approximations (MPA) were applied to power series solutions of Cauchy problems
that solved by using He’s variational iteration method (VIM). Then, numerical
results obtained by using multivariate Padé approximations were compared with
the exact solutions of Cauchy problems.

Keywords

Cauchy problem, inviscid Burger’s equation

References

• N. Guzel, M. Bayram, On the numerical solution of differential-algebraic equations with index-3, Applied Mathematics and Computation (2006), 175(2), 1320-1331. E. Celik, M. Bayram, http://www.sciencedirect.com/science/article/pii/S0096300303007197Numerical solution of differential–algebraic equation systems and applications, Applied Mathematics and Computation (2004), 154 ( 2): 405-413.
• V. Turut and N Guzel., Comparing Numerical Methods for Solving Time- Fractional Reaction-Diffusion Equations, ISRN Mathematical Analysis (2012), Doi:10.5402/2012/737206.
• V. Turut, N. Güzel, Multivariate padé approximation for solving partial differential equations of fractional order”,Abstract and Applied Analysis (2013), in press.
• V. Turut, E. Çelik, M. Yiğider, Multivariate padé approximation for solving partial differential equations (PDE), International Journal For Numerical Methods In Fluids (2011), 66 (9):1159-1173. X. W. Zhou, L. Yao, http://www.sciencedirect.com/science/article/pii/S0898122110003718The variational iteration method for Cauchy problems, Computers & Mathematics with Applications (2010), 60 ( 3): 756-760. J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. (1998), 167: 57-68.
• J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non-Linear. Mech. (1999), 34: 699-708. J.H. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. (2007), 207: 3-17. J.H. He, X.H. Wu, Variational iteration method: new development and applications, Comput. Math. Appl. (2007), 54: 881-894. J.H. He, G.-C. Wu, F. Austin, The variational iteration method which should be followed, Nonlinear Sci. Lett. A (2010), 1: 1-30.
• A. Cuyt, L. Wuytack, Nonlinear Methods in Numerical Analysis, Elsevier Science Publishers B.V. (1987), Amsterdam Abdul-Majid Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press (2009), Beijing G. Baker , P. Graves-Morris ,Padé Approximants. Basic Theory. Encyclopedia of Mathematics and its applications: vol 13., Addison- Wpsley, Reading (1981), Massachusetts.
• A. Cuyt, A multivariate convergence theorem of the “de Montessus de Ballore” type, J. Comput. Appl. Math. (1990), 32: 47-57.
• A. Cuyt, K. Driver and D.S. Lubinsky, Nuttall-Pommerenke theorem for homogeneous Padé approximants, J. Comput. Appl. Math. (1996), 67: 141-146. A. Cuyt, K. Driver and D.S. Lubinsky, A direct approach to convergence of multivariate, non-homogeneous, Padé approximants, J. Comput. Appl. Math. (1996), 69: 353-366.
• C. Brezinski, Extrapolation algorithms and Padé approximations: a historical survey, Appl. Numer. Math. (1996), 20: 299-318.