Numerical Solutions of the Regularized Long-Wave (RLW) Equation Using New Modification of Laplace-Decomposition Method

Affiliation(s)

Department of Mathematics, Science Faculty for Girls, King Abdulaziz University, Jeddah, KSA.

Department of Mathematics, Science Faculty for Girls, King Abdulaziz University, Jeddah, KSA.

ABSTRACT

In this paper the new modification of Laplace Adomian decomposition method (ADM) to obtain numerical solution of the regularized long-wave (RLW) equation is presented. The performance of the method is illustrated by solving two test examples of the problem. To see the accuracy of the method, *L*_{2} and *L*_{∞} error norms are calculated.

Cite this paper

N. Al-Zaid, H. Bakodah and F. Hendi, "Numerical Solutions of the Regularized Long-Wave (RLW) Equation Using New Modification of Laplace-Decomposition Method,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 159-163. doi: 10.4236/apm.2013.31A022.

N. Al-Zaid, H. Bakodah and F. Hendi, "Numerical Solutions of the Regularized Long-Wave (RLW) Equation Using New Modification of Laplace-Decomposition Method,"

References

[1] D. H. Peregrine, “Calculation of the Development of an Undular Bore,” Journal of Fluid Mechanics, Vol. 25, No. 2, 1966, pp. 321-330. doi:10.1017/S0022112066001678

[2] Kh. O. Abdullove, H. Bogalubsky and V. G. Markhankov, “One More Example of Inelastic Soliton Interaction,” Physics Letters A, Vol. 56, No. 6, 1976, pp. 427-428.

[3] J. L. Bona, W. G. Pritchard and L. R. Scott, “Numerical Schemes for a Model of Nonlinear Dispersive Waves,” Journal of Computer Physics, Vol. 60, 1985, pp. 167-196.

[4] P. J. Jain and L. Iskandar, “Numerical Solutions of the Regularized Long Wave Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 20, No. 2, 1979, pp. 195-201. doi:10.1016/0045-7825(79)90017-3

[5] L. R. T. Gardner, G. A. Gardner and A. Dogan, “A Least Squares Finite Element Scheme for RLW Equation,” Communications in Numerical Methods in Engineering, Vol. 11, No. 1, 1995, pp. 59-68. doi:10.1002/cnm.1640110109

[6] I. Da□ and M. N. Ozer, “Approximation of the RLW Equation by the Least Square Cubic B-Spline Finite Element Method,” Applied Mathematical Modelling, Vol. 25, No. 3, 2001, pp. 221-231. doi:10.1016/S0307-904X(00)00030-5

[7] Y. keskin and G. Oturanc, “Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method,” Applied Mathematical Sciences, Vol. 4, No. 25, 2010, pp. 1221-1231.

[8] G. Adomian, “Nonlinear Stochastic Operator Equataions,” Academic Press, San Diego, 1986.

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

[10] E. Yusufoglu and A. Bekir, “Application of the Variational Iteration Method to the Regularized Long Wave Equation,” Computer & Mathematics with Applications, Vol. 25, 2003, pp. 321-330.

[11] G. Adomian, “Modified Adomian Polynomials,” Mathematical Computation, Vol. 76, No. 1, 1996, pp. 95-97. doi:10.1016/0096-3003(95)00186-7

[12] A.-M. Wazwaz and S. M. El-Sayed, “A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators,” Applied Mathematics and Computation, Vol. 122, No. 3, 2001, pp. 393-405. doi:10.1016/S0096-3003(00)00060-6

[13] T. S. El-Danaf, M. A. Ramadan and F. E. I. AbdAlaal, “The Use of Adomian Decomposition Method for Solving the Regularized Long-Wave Equation,” Chaos, Solitons & Fractals, Vol. 26, No. 3, 2005, pp. 747-757. doi:10.1016/j.chaos.2005.02.012

[14] D. Kaya and S. M. El-Sayed, “An Application of the Decomposition Method for the Generalized KdV and RLW Equations,” Chaos, Solitons & Fractals, Vol. 17, No 5, 2003, pp. 869-877. doi:10.1016/S0960-0779(02)00569-6

[1] D. H. Peregrine, “Calculation of the Development of an Undular Bore,” Journal of Fluid Mechanics, Vol. 25, No. 2, 1966, pp. 321-330. doi:10.1017/S0022112066001678

[2] Kh. O. Abdullove, H. Bogalubsky and V. G. Markhankov, “One More Example of Inelastic Soliton Interaction,” Physics Letters A, Vol. 56, No. 6, 1976, pp. 427-428.

[3] J. L. Bona, W. G. Pritchard and L. R. Scott, “Numerical Schemes for a Model of Nonlinear Dispersive Waves,” Journal of Computer Physics, Vol. 60, 1985, pp. 167-196.

[4] P. J. Jain and L. Iskandar, “Numerical Solutions of the Regularized Long Wave Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 20, No. 2, 1979, pp. 195-201. doi:10.1016/0045-7825(79)90017-3

[5] L. R. T. Gardner, G. A. Gardner and A. Dogan, “A Least Squares Finite Element Scheme for RLW Equation,” Communications in Numerical Methods in Engineering, Vol. 11, No. 1, 1995, pp. 59-68. doi:10.1002/cnm.1640110109

[6] I. Da□ and M. N. Ozer, “Approximation of the RLW Equation by the Least Square Cubic B-Spline Finite Element Method,” Applied Mathematical Modelling, Vol. 25, No. 3, 2001, pp. 221-231. doi:10.1016/S0307-904X(00)00030-5

[7] Y. keskin and G. Oturanc, “Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method,” Applied Mathematical Sciences, Vol. 4, No. 25, 2010, pp. 1221-1231.

[8] G. Adomian, “Nonlinear Stochastic Operator Equataions,” Academic Press, San Diego, 1986.

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

[10] E. Yusufoglu and A. Bekir, “Application of the Variational Iteration Method to the Regularized Long Wave Equation,” Computer & Mathematics with Applications, Vol. 25, 2003, pp. 321-330.

[11] G. Adomian, “Modified Adomian Polynomials,” Mathematical Computation, Vol. 76, No. 1, 1996, pp. 95-97. doi:10.1016/0096-3003(95)00186-7

[12] A.-M. Wazwaz and S. M. El-Sayed, “A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators,” Applied Mathematics and Computation, Vol. 122, No. 3, 2001, pp. 393-405. doi:10.1016/S0096-3003(00)00060-6

[13] T. S. El-Danaf, M. A. Ramadan and F. E. I. AbdAlaal, “The Use of Adomian Decomposition Method for Solving the Regularized Long-Wave Equation,” Chaos, Solitons & Fractals, Vol. 26, No. 3, 2005, pp. 747-757. doi:10.1016/j.chaos.2005.02.012

[14] D. Kaya and S. M. El-Sayed, “An Application of the Decomposition Method for the Generalized KdV and RLW Equations,” Chaos, Solitons & Fractals, Vol. 17, No 5, 2003, pp. 869-877. doi:10.1016/S0960-0779(02)00569-6