A new algorithm for solving nonlinear boundary value problems

Purpose – The purpose of this paper is to present a new algorithm for solving nonlinear boundary value problems (BVPs).Design/methodology/approach – The method converts the nonlinear BVP into a system of nonlinear equations. By solving the system, the solution can be determined. Comparing the methodology with some known techniques shows that the present approach is simple, easy to use and highly accurate.Findings – The proposed technique allows us to obtain an approximate solution in a series form which satisfies all the given conditions. Test problems are given to illustrate the pertinent features of the method. The accuracy of the numerical results indicates that the technique is efficient and well suited for solving nonlinear differential equations with initial and boundary conditions.Originality/value – The paper provides a reliable technique which avoids the tedious work needed by classical techniques and existing numerical methods and does not require discretization in order to find the solutions of...

[1]  Lionel Gabet.  Modélisation de la diffusion de médicaments à travers les capillaires et dans les tissus à la suite d'une injection et esquisse d'une théorie décompositionnelle et application aux équations aux dérivées partielles , 1992 .

[2]  M. L. Krasnov,et al.  Recueil de problèmes sur les équations différentielles ordinaires , 1981 .

[3]  G. Adomian,et al.  A modified decomposition , 1992 .

[4]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[5]  Feyed Ben Zitoun,et al.  A method for solving nonlinear differential equations , 2010, Kybernetes.

[6]  Feyed Ben Zitoun,et al.  A Taylor expansion approach using Faà di Bruno's formula for solving nonlinear integral equations of the second and third kind , 2009, Kybernetes.

[7]  Warren P. Johnson.  The Curious History of Faà di Bruno's Formula , 2002, Am. Math. Mon..

[8]  L. Gabet,et al.  The theoretical foundation of the Adomian method , 1994 .

[9]  Y Cherruault,et al.  Practical formulae for calculation of Adomian's polynomials and application to the convergence of the decomposition method. , 1994, International journal of bio-medical computing.

[10]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

[11]  Mehmet Sezer,et al.  On the solution of the Riccati equation by the Taylor matrix method , 2006, Appl. Math. Comput..

[12]  Y. Cherruault.  A comparison of numerical solutions of fourth‐order boundary value problems , 2005 .

[13]  Ángel Martín del Rey,et al.  Faa' di Bruno's formula, lattices, and partitions , 2005, Discret. Appl. Math..

[14]  R. Temam,et al.  Inertial manifolds and slow manifolds , 1991 .

[15]  Eugene M. Klimko,et al.  An algorithm for calculating indices in Fàa di Bruno's formula , 1973 .

[16]  Mehmet Sezer,et al.  A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials , 1996 .

[17]  David J. Evans,et al.  Collocation approximation for fourth-order boundary value problems , 1997, Int. J. Comput. Math..

[18]  J. Baxley.  Nonlinear Two Point Boundary Value Problems , 1968 .

[19]  A. Beardon.  The Differentiation of a Composite Function , 1971, The Mathematical Gazette.

[20]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[21]  J. Riordan.  Derivatives of composite functions , 1946 .

[22]  D. M. Hutton,et al.  Modèles et méthodes mathématiques pour les sciences du vivant , 1998 .

[23]  Esquisse d'une théorie décompositionnelle , 1996 .

[24]  D. M. Hutton,et al.  Optimisation - Mèthodes locales et globales , 1999 .

[25]  Ernst Hairer.  Analyse II (Calcul Différentiel et Equations Différentielles) , 1999 .

[26]  Y. Cherruault,et al.  Convergence of Adomian's method applied to differential equations , 1994 .

[27]  A. Wazwaz.  Approximate solutions to boundary value problems of higher order by the modified decomposition method , 2000 .

[28]  Abdul-Majid Wazwaz,et al.  The Numerical Solution of Special Fourth-Order Boundary Value Problems by the Modified Decomposition Method , 2000, Neural Parallel Sci. Comput..

[29]  George Adomian,et al.  Transformation of series , 1991 .

[30]  Ravi P. Agarwal,et al.  Boundary value problems for higher order differential equations , 1986 .

[31]  D. O’Regan.  Existence Theory for Nonlinear Ordinary Differential Equations , 1997 .