Modified Adomian Techniques Applied to Non-Linear Volterra Integral Equations

Affiliation(s)

Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya.

Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya.

ABSTRACT

In this work, we studied the performance of modified techniques of
Adomian method applied to non-linear Volterra integral equations of the
second kind. This study shows that the modified techniques are reliable,
efficient and easy to use through recursive relations that involve simple
integrals. Furthermore, we found that the right choice and the proper
implementation of the modified techniques reduce the computational difficulties
and increase the speed of convergent, comparing with the standard Adomian method.

Cite this paper

H. Ali and F. Abdelwahid, "Modified Adomian Techniques Applied to Non-Linear Volterra Integral Equations,"*Open Journal of Applied Sciences*, Vol. 3 No. 2, 2013, pp. 202-207. doi: 10.4236/ojapps.2013.32027.

H. Ali and F. Abdelwahid, "Modified Adomian Techniques Applied to Non-Linear Volterra Integral Equations,"

References

[1] H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 235-247. doi:10.1016/0168-9274(92)90018-9

[2] T. Tang, S. McKee and T. Diogo, “Product Integration Methods for an Integral Equation with Logarithmic Singular Kernel,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 259-266. doi:10.1016/0168-9274(92)90020-E

[3] T. Diogo, S. McKee and T. Tang, “A Hermite-Type Col location Method for the Solution of an Integral Equation with a Certain Weakly Singular Kernel,” IMA Journal of Numerical Analysis, Vol. 11, No. 4, 1991, pp. 595-605. doi:10.1093/imanum/11.4.595

[4] G. Adomian, “The Decomposition Method for Nonlinear Dynamical Systems,” Journal of Mathematical Analysis and Applications, Vol. 120, No. 1, 1986, pp. 370-383. doi:10.1016/0022-247X(86)90223-4

[5] G. Adomian, “A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations,” Mathematical and Computer Modeling, Vol. 13, No. 7, 1990, pp. 17-43. doi:10.1016/0895-7177(90)90125-7

[6] F. Abdelwahid, “A Mathematical Model of Adomian Polynomials,” Applied Mathematics and Computation, Vol. 141, No. 2-3, 2003, pp. 447-453. doi:10.1016/S0096-3003(02)00266-7

[7] F. Abdelwahid, “Adomian Decomposition Method Applied to Nonlinear Integral Equations,” Alexandria Journal of Mathematics, Vol. 1, No. 1, 2010, pp. 11-18.

[8] A. Wazwaz, “A Reliable Modification of Adomian De composition Method,” Applied Mathematics and Computation, Vol. 102, No. 1, 1999, pp. 77-86. doi:10.1016/S0096-3003(98)10024-3

[9] M. Hosseini, “Adomian Decomposition Method with Chebyshev Polynomials,” Applied Mathematics and Computation, Vol. 175, No. 2, 2006, pp. 1685-1693. doi:10.1016/j.amc.2005.09.014

[1] H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 235-247. doi:10.1016/0168-9274(92)90018-9

[2] T. Tang, S. McKee and T. Diogo, “Product Integration Methods for an Integral Equation with Logarithmic Singular Kernel,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 259-266. doi:10.1016/0168-9274(92)90020-E

[3] T. Diogo, S. McKee and T. Tang, “A Hermite-Type Col location Method for the Solution of an Integral Equation with a Certain Weakly Singular Kernel,” IMA Journal of Numerical Analysis, Vol. 11, No. 4, 1991, pp. 595-605. doi:10.1093/imanum/11.4.595

[4] G. Adomian, “The Decomposition Method for Nonlinear Dynamical Systems,” Journal of Mathematical Analysis and Applications, Vol. 120, No. 1, 1986, pp. 370-383. doi:10.1016/0022-247X(86)90223-4

[5] G. Adomian, “A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations,” Mathematical and Computer Modeling, Vol. 13, No. 7, 1990, pp. 17-43. doi:10.1016/0895-7177(90)90125-7

[6] F. Abdelwahid, “A Mathematical Model of Adomian Polynomials,” Applied Mathematics and Computation, Vol. 141, No. 2-3, 2003, pp. 447-453. doi:10.1016/S0096-3003(02)00266-7

[7] F. Abdelwahid, “Adomian Decomposition Method Applied to Nonlinear Integral Equations,” Alexandria Journal of Mathematics, Vol. 1, No. 1, 2010, pp. 11-18.

[8] A. Wazwaz, “A Reliable Modification of Adomian De composition Method,” Applied Mathematics and Computation, Vol. 102, No. 1, 1999, pp. 77-86. doi:10.1016/S0096-3003(98)10024-3

[9] M. Hosseini, “Adomian Decomposition Method with Chebyshev Polynomials,” Applied Mathematics and Computation, Vol. 175, No. 2, 2006, pp. 1685-1693. doi:10.1016/j.amc.2005.09.014