Adomian Decomposition Method Associated with Boole-s Integration Rule for Goursat Problem
The Goursat partial differential equation arises in
linear and non linear partial differential equations with mixed
derivatives. This equation is a second order hyperbolic partial
differential equation which occurs in various fields of study such as
in engineering, physics, and applied mathematics. There are many
approaches that have been suggested to approximate the solution of
the Goursat partial differential equation. However, all of the
suggested methods traditionally focused on numerical differentiation
approaches including forward and central differences in deriving the
scheme. An innovation has been done in deriving the Goursat partial
differential equation scheme which involves numerical integration
techniques. In this paper we have developed a new scheme to solve
the Goursat partial differential equation based on the Adomian
decomposition (ADM) and associated with Boole-s integration rule to
approximate the integration terms. The new scheme can easily be
applied to many linear and non linear Goursat partial differential
equations and is capable to reduce the size of computational work.
The accuracy of the results reveals the advantage of this new scheme
over existing numerical method.
[1] G. Adomian, A Review of the Decomposition Method in Applied
Mathematics, Journal of Mathematical Analysis and Application,
Vol.135, No.2, 1988, pp.501-544.
[2] M.A. A1-Alaoui, A Class of Numerical Integration Rule With First
Class Order Derivatives, University Research Board of the American
University of Beirut, 1990, pp.25-44.
[3] Y.L. An and W.C. Hua, The Discontinuous Ivp of A Reacting Gas
Flow System, Transaction of the American Mathematical Society,
1981.
[4] R.K. Arnold and W.U. Christoph, Numerical Integration on Advanced
Computer System, Germany: Springer-Verlag Berlin Heidelberg, 1994,
pp.5-23.
[5] S.A. Aseeri, Goursat Functions for A Problem of An Isotropic Plate
with A Curvilinear Hole, International Journal for Open Problems
Computer Mathematic, Vol.1, 2008, pp.266-285.
[6] L.G. Bushnell, D. Tilbury and S.S. Sastry, Extended Goursat Normal
Form With Application to Nonholonomic Motion Planning, Electronic
Research Laboratory University of California, 1994, pp.1-12.
[7] S.M. Cathleen, The Mathematical Approach to the Sonic Barrier,
Bulletin of the American Mathematical Society, Vol.2, 1982, pp.127-
145.
[8] S. Chen and D. Li, Supersonic Flow Past A Symmetrically Curved
Cone, Indiana University Mathematics Journal, 2000.
[9] T. Dawn, Trajectory Generation for the N-Trailer Problem Using
Goursat Normal Form, IEEE Transactions on Automatic Control,
Vol.40, 1995, pp.802-819.
[10] A.I. Ismail and M.A.S. Nasir, Numerical Solution of the Goursat
Problem, IASTED Conference on Applied Simulation And Modelling,
2004, pp.243-246.
[11] Y. Susumu, Goursat Problem for A Microdifferential Operator of
Fuchsian Type and Its Application, RIMS Kyoto University, Vol.33,
1997, pp.559-641.
[12] H. Taghvafard and H.G. Erjaee, Two-Dimensional Differential
Transform Method for Solving Linear and Non-Linear Goursat
problem, International Journal for Engineering and Mathematical
Sciences, Vol.6, No.2, 2010, pp.103-106.
[13] A.M. Wazwaz, On the Numerical Solution for the Goursat Problem,
Applied Mathematics and Computational, Vol.59, 1993, pp.89-95.
[14] A.M. Wazwaz, The Decomposition Method for Approximate Solution
of Goursat Problem, Applied Mathematics and Computational, Vol.69,
1995, pp.229-311.
[1] G. Adomian, A Review of the Decomposition Method in Applied
Mathematics, Journal of Mathematical Analysis and Application,
Vol.135, No.2, 1988, pp.501-544.
[2] M.A. A1-Alaoui, A Class of Numerical Integration Rule With First
Class Order Derivatives, University Research Board of the American
University of Beirut, 1990, pp.25-44.
[3] Y.L. An and W.C. Hua, The Discontinuous Ivp of A Reacting Gas
Flow System, Transaction of the American Mathematical Society,
1981.
[4] R.K. Arnold and W.U. Christoph, Numerical Integration on Advanced
Computer System, Germany: Springer-Verlag Berlin Heidelberg, 1994,
pp.5-23.
[5] S.A. Aseeri, Goursat Functions for A Problem of An Isotropic Plate
with A Curvilinear Hole, International Journal for Open Problems
Computer Mathematic, Vol.1, 2008, pp.266-285.
[6] L.G. Bushnell, D. Tilbury and S.S. Sastry, Extended Goursat Normal
Form With Application to Nonholonomic Motion Planning, Electronic
Research Laboratory University of California, 1994, pp.1-12.
[7] S.M. Cathleen, The Mathematical Approach to the Sonic Barrier,
Bulletin of the American Mathematical Society, Vol.2, 1982, pp.127-
145.
[8] S. Chen and D. Li, Supersonic Flow Past A Symmetrically Curved
Cone, Indiana University Mathematics Journal, 2000.
[9] T. Dawn, Trajectory Generation for the N-Trailer Problem Using
Goursat Normal Form, IEEE Transactions on Automatic Control,
Vol.40, 1995, pp.802-819.
[10] A.I. Ismail and M.A.S. Nasir, Numerical Solution of the Goursat
Problem, IASTED Conference on Applied Simulation And Modelling,
2004, pp.243-246.
[11] Y. Susumu, Goursat Problem for A Microdifferential Operator of
Fuchsian Type and Its Application, RIMS Kyoto University, Vol.33,
1997, pp.559-641.
[12] H. Taghvafard and H.G. Erjaee, Two-Dimensional Differential
Transform Method for Solving Linear and Non-Linear Goursat
problem, International Journal for Engineering and Mathematical
Sciences, Vol.6, No.2, 2010, pp.103-106.
[13] A.M. Wazwaz, On the Numerical Solution for the Goursat Problem,
Applied Mathematics and Computational, Vol.59, 1993, pp.89-95.
[14] A.M. Wazwaz, The Decomposition Method for Approximate Solution
of Goursat Problem, Applied Mathematics and Computational, Vol.69,
1995, pp.229-311.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58841", author = "Mohd Agos Salim Nasir and Ros Fadilah Deraman and Siti Salmah Yasiran", title = "Adomian Decomposition Method Associated with Boole-s Integration Rule for Goursat Problem", abstract = "The Goursat partial differential equation arises in
linear and non linear partial differential equations with mixed
derivatives. This equation is a second order hyperbolic partial
differential equation which occurs in various fields of study such as
in engineering, physics, and applied mathematics. There are many
approaches that have been suggested to approximate the solution of
the Goursat partial differential equation. However, all of the
suggested methods traditionally focused on numerical differentiation
approaches including forward and central differences in deriving the
scheme. An innovation has been done in deriving the Goursat partial
differential equation scheme which involves numerical integration
techniques. In this paper we have developed a new scheme to solve
the Goursat partial differential equation based on the Adomian
decomposition (ADM) and associated with Boole-s integration rule to
approximate the integration terms. The new scheme can easily be
applied to many linear and non linear Goursat partial differential
equations and is capable to reduce the size of computational work.
The accuracy of the results reveals the advantage of this new scheme
over existing numerical method.", keywords = "Goursat problem, partial differential equation,
Adomian decomposition method, Boole's integration rule.", volume = "6", number = "12", pages = "1726-5", }