Adomian Decomposition Method for Solving Goursat's Problems

Author(s)
Mariam A. Al-Mazmumy

ABSTRACT

In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.

In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.

KEYWORDS

Goursat’s Problem, Linear and Nonlinear Hyperbolic Equation of Second and Fourth-Orders, System of Linear Hyperbolic Equations of Second Order, Adomian Decomposition Method

Goursat’s Problem, Linear and Nonlinear Hyperbolic Equation of Second and Fourth-Orders, System of Linear Hyperbolic Equations of Second Order, Adomian Decomposition Method

Cite this paper

nullM. Al-Mazmumy, "Adomian Decomposition Method for Solving Goursat's Problems,"*Applied Mathematics*, Vol. 2 No. 8, 2011, pp. 975-980. doi: 10.4236/am.2011.28134.

nullM. Al-Mazmumy, "Adomian Decomposition Method for Solving Goursat's Problems,"

References

[1] E. Goursat, “A Course in Mathematical Analysis, Vol. 3: Variation of Solutions and Partial Differential Equations of the Second Order and Integral Equations and Calculus of Variations,” Gauthier-Villars, Paris, 1923.

[2] G. Adomian, “Nonlinear Stochastic Operator Equations,” Academic Press, Orlando, 1986.

[3] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994.

[4] G. Adomian and R. Rach, “Transformation of Series,” Applied Mathematics Letters, Vol. 4, No. 4, 1991, pp. 69-71. doi:10.1016/0893-9659(91)90058-4

[5] G. Adomian, R. Rach and R. E. Meyers, “A Modified Decomposition,” Computers & Mathematics with Applications, Vol. 23, No. 1, January 1992, pp. 17-23. doi:10.1016/0898-1221(92)90076-T

[6] G. Adomian and R. Rach, “Inhomogeneous Nonlinear Partial Differential Equations with Variable Coefficients,” Applied Mathematics Letters, Vol. 5, No. 2, March 1992, pp. 11-12. doi:10.1016/0893-9659(92)90101-E

[7] G. Adomian and R. Rach, “Nonlinear Transformation of Series Part II,” Computers & Mathematics with Applications, Vol. 23, No. 10, May 1992, pp. 79-83. doi:10.1016/0898-1221(92)90058-P

[8] G. Adomian and R. Rach, “Modified Decomposition Solution of Nonlinear Partial Differential Equations,” Applied Mathematics Letters, Vol. 5, No. 6, November 1992, pp. 29-30. doi:10.1016/0893-9659(92)90008-W

[9] G. Adomian and R. Rach, “Solution of Nonlinear Partial Differential Equations in One, Two, Three, and four Dimensions,” World Scientific Series in Applicable Analysis, Vol. 2, 1993, pp. 1-13.

[10] G. Adomian and R. Rach, “Modified Decomposition Solution of Linear and Nonlinear Boundary-Value Problems,” Nonlinear Analysis, Vol. 23, No. 5, September 1994, pp. 615-619. doi:10.1016/0362-546X(94)90240-2

[11] G. Adomian and R. Rach, “Analytic Solution of Nonlinear Boundary-Value Problems in Several Dimensions by Decomposition,” Journal of Mathematical Analysis and Applications, Vol. 174, No. 1, 15 March 1993, pp. 118-137. doi:10.1006/jmaa.1993.1105

[12] G. Adomian and R. Rach, “A New Algorithm for Matching Boundary Conditions in Decomposition Solutions,” Applied Mathematics and Computation, Vol. 58, No. 1, September 1993, pp. 61-68. doi:10.1016/0096-3003(93)90012-4

[13] A. Wazwaz, “The Decomposition Method for Approximate Solution of the Goursat Problem,” Applied Mathematics and Computation, Vol. 69, No. 2-3, May 1995, pp. 299-311. doi:10.1016/0096-3003(94)00137-S

[1] E. Goursat, “A Course in Mathematical Analysis, Vol. 3: Variation of Solutions and Partial Differential Equations of the Second Order and Integral Equations and Calculus of Variations,” Gauthier-Villars, Paris, 1923.

[2] G. Adomian, “Nonlinear Stochastic Operator Equations,” Academic Press, Orlando, 1986.

[3] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994.

[4] G. Adomian and R. Rach, “Transformation of Series,” Applied Mathematics Letters, Vol. 4, No. 4, 1991, pp. 69-71. doi:10.1016/0893-9659(91)90058-4

[5] G. Adomian, R. Rach and R. E. Meyers, “A Modified Decomposition,” Computers & Mathematics with Applications, Vol. 23, No. 1, January 1992, pp. 17-23. doi:10.1016/0898-1221(92)90076-T

[6] G. Adomian and R. Rach, “Inhomogeneous Nonlinear Partial Differential Equations with Variable Coefficients,” Applied Mathematics Letters, Vol. 5, No. 2, March 1992, pp. 11-12. doi:10.1016/0893-9659(92)90101-E

[7] G. Adomian and R. Rach, “Nonlinear Transformation of Series Part II,” Computers & Mathematics with Applications, Vol. 23, No. 10, May 1992, pp. 79-83. doi:10.1016/0898-1221(92)90058-P

[8] G. Adomian and R. Rach, “Modified Decomposition Solution of Nonlinear Partial Differential Equations,” Applied Mathematics Letters, Vol. 5, No. 6, November 1992, pp. 29-30. doi:10.1016/0893-9659(92)90008-W

[9] G. Adomian and R. Rach, “Solution of Nonlinear Partial Differential Equations in One, Two, Three, and four Dimensions,” World Scientific Series in Applicable Analysis, Vol. 2, 1993, pp. 1-13.

[10] G. Adomian and R. Rach, “Modified Decomposition Solution of Linear and Nonlinear Boundary-Value Problems,” Nonlinear Analysis, Vol. 23, No. 5, September 1994, pp. 615-619. doi:10.1016/0362-546X(94)90240-2

[11] G. Adomian and R. Rach, “Analytic Solution of Nonlinear Boundary-Value Problems in Several Dimensions by Decomposition,” Journal of Mathematical Analysis and Applications, Vol. 174, No. 1, 15 March 1993, pp. 118-137. doi:10.1006/jmaa.1993.1105

[12] G. Adomian and R. Rach, “A New Algorithm for Matching Boundary Conditions in Decomposition Solutions,” Applied Mathematics and Computation, Vol. 58, No. 1, September 1993, pp. 61-68. doi:10.1016/0096-3003(93)90012-4

[13] A. Wazwaz, “The Decomposition Method for Approximate Solution of the Goursat Problem,” Applied Mathematics and Computation, Vol. 69, No. 2-3, May 1995, pp. 299-311. doi:10.1016/0096-3003(94)00137-S