Solving Nonlinear Stochastic Diffusion Models with Nonlinear Losses Using the Homotopy Analysis Method

Author(s)
Aisha A. Fareed,
Hanafy H. El-Zoheiry,
Magdy A. El-Tawil,
Mohammed A. El-Beltagy,
Hany N. Hassan

Affiliation(s)

Department of Basic Sciences, Engineering Faculty, Benha University, Benha, Egypt.

Department of Engineering Mathematics & Physics, Engineering Faculty, Cairo University, Cairo, Egypt.

Department of Electrical & Computer Engineering, Engineering Faculty, Effat University, Jeddah, KSA.

Department of Basic Sciences, Engineering Faculty, Benha University, Benha, Egypt.

Department of Engineering Mathematics & Physics, Engineering Faculty, Cairo University, Cairo, Egypt.

Department of Electrical & Computer Engineering, Engineering Faculty, Effat University, Jeddah, KSA.

ABSTRACT

This paper deals with the construction of approximate series solutions of diffusion models with stochastic excitation and nonlinear losses using the homotopy analysis method (HAM). The mean, variance and other statistical properties of the stochastic solution are computed. The solution technique was applied successfully to the 1D and 2D diffusion models. The scheme shows importance of choice of convergence-control parameter to guarantee the convergence of the solutions of nonlinear differential Equations. The results are compared with the Wiener-Hermite expansion with perturbation (WHEP) technique and good agreements are obtained.

Cite this paper

A. Fareed, H. El-Zoheiry, M. El-Tawil, M. El-Beltagy and H. Hassan, "Solving Nonlinear Stochastic Diffusion Models with Nonlinear Losses Using the Homotopy Analysis Method,"*Applied Mathematics*, Vol. 5 No. 1, 2014, pp. 115-127. doi: 10.4236/am.2014.51014.

A. Fareed, H. El-Zoheiry, M. El-Tawil, M. El-Beltagy and H. Hassan, "Solving Nonlinear Stochastic Diffusion Models with Nonlinear Losses Using the Homotopy Analysis Method,"

References

[1] M. A. El-Tawil and A. S. Al-Jihany, “On the Solution of Stochastic Oscillatory Quadratic Nonlinear Equations Using Different Techniques, a Comparison Study,” Topological Methods in Nonlinear Analysis, Vol. 31, No. 2, 2008, pp. 315-330.

[2] M. A. El-Tawil and N. A. Al-Mulla, “Using Homotopy WHEP Technique for Solving A Stochastic Nonlinear Diffusion Equation,” Mathematical and Computer Modelling, Vol. 51, No. 9, 2010, pp. 1277-1284.

[3] J. C. Cortes, J. V. Romero, M. D. Rosello and C. Santamaria, “Solving Random Diffusion Models with Nonlinear Perturbations by the Wiener-Hermite Expansion 617 Method,” Computers & Mathematics with Applications, Vol. 61, No. 8, 2011, pp. 1946-1950.

[4] C. Q. Dai and J. F. Zhang, “Application of He’s Exp-Function Method to the Stochastic mKdV Equation,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, No. 5, 2009, pp. 675-680.

[5] M. El-Beltagy and M. El-Tawil, “Toward a Solution of a Class of Non-Linear Stochastic Perturbed PDEs Using Automated WHEP Algorithm,” Applied Mathematical Modeling, Vol. 37, No. 12-13, 2013, pp. 7174-7192.

http://dx.doi.org/10.1016/j.apm.2013.01.038

[6] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[7] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theories,” Communications in Nonlinear Science Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997.

[8] G. Adomian, “A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations,” Computers and Mathematics with Applications, Vol. 21, No. 5, 1991, pp. 101-127.

[9] J. H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79.

[10] T. Hayat and M. Sajid, “Analytic Solution for Axisymmetric Flow and Heat Transfer of a Second Grade Fluid Past a Stretching Sheet,” International Journal of Heat and Mass Transfer, Vol. 50, No. 1-2, 2007, pp. 75-84.

[11] S. Abbasbandy, “Soliton Solutions for the 5th-Order KdV Equation with the Homotopy Analysis Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 83-87.

[12] Y. P. Liu and Z. B. Li, “The Homotopy Analysis Method for Approximating the Solution of the Modified Korteweg-de Vries Equation,” Chaos, Solitons and Fractals, Vol. 39, No. 1, 2009, pp. 1-8.

[13] Y. Bouremel, “Explicit Series Solution for the Glauert-Jet Problem by Means of the Homotopy Analysis Method,” Communication in Nonlinear Science Numerical Simulation, Vol. 12, No. 5, 2007, pp. 714-724.

[14] A. Molabahrami and F. Khani, “The Homotopy Analysis Method to Solve the Burgers-Huxley Equation,” Nonlinear Analysis Real World Applications, Vol. 10, No. 2, 2009, pp. 589-600.

[15] S. Abbasbandy, E. Magyari and E. Shivanian, “The Homotopy Analysis Method for Multiple Solutions of Nonlinear Boundary Value Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 9-10, 2009, pp. 3530-3536.

[16] H. N. Hassan and M. A. El-Tawil, “Solving Cubic and Coupled Nonlinear Schrodinger Equations Using the Homotopy Analysis Method,” International Journal of Applied Mathematics and Mechanics, Vol. 7, No. 8, 2011, pp. 41-64.

[17] H. N. Hassan and M. A. El-Tawil, “An Efficient Analytic Approach for Solving Two-Point Nonlinear Boundary Value Problems by Homotopy Analysis Method,” Mathematical Methods in the Applied Sciences, Vol. 34, No. 8, 2011, pp. 977-989.

[1] M. A. El-Tawil and A. S. Al-Jihany, “On the Solution of Stochastic Oscillatory Quadratic Nonlinear Equations Using Different Techniques, a Comparison Study,” Topological Methods in Nonlinear Analysis, Vol. 31, No. 2, 2008, pp. 315-330.

[2] M. A. El-Tawil and N. A. Al-Mulla, “Using Homotopy WHEP Technique for Solving A Stochastic Nonlinear Diffusion Equation,” Mathematical and Computer Modelling, Vol. 51, No. 9, 2010, pp. 1277-1284.

[3] J. C. Cortes, J. V. Romero, M. D. Rosello and C. Santamaria, “Solving Random Diffusion Models with Nonlinear Perturbations by the Wiener-Hermite Expansion 617 Method,” Computers & Mathematics with Applications, Vol. 61, No. 8, 2011, pp. 1946-1950.

[4] C. Q. Dai and J. F. Zhang, “Application of He’s Exp-Function Method to the Stochastic mKdV Equation,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, No. 5, 2009, pp. 675-680.

[5] M. El-Beltagy and M. El-Tawil, “Toward a Solution of a Class of Non-Linear Stochastic Perturbed PDEs Using Automated WHEP Algorithm,” Applied Mathematical Modeling, Vol. 37, No. 12-13, 2013, pp. 7174-7192.

http://dx.doi.org/10.1016/j.apm.2013.01.038

[6] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[7] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theories,” Communications in Nonlinear Science Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997.

[8] G. Adomian, “A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations,” Computers and Mathematics with Applications, Vol. 21, No. 5, 1991, pp. 101-127.

[9] J. H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79.

[10] T. Hayat and M. Sajid, “Analytic Solution for Axisymmetric Flow and Heat Transfer of a Second Grade Fluid Past a Stretching Sheet,” International Journal of Heat and Mass Transfer, Vol. 50, No. 1-2, 2007, pp. 75-84.

[11] S. Abbasbandy, “Soliton Solutions for the 5th-Order KdV Equation with the Homotopy Analysis Method,” Nonlinear Dynamics, Vol. 51, No. 1-2, 2008, pp. 83-87.

[12] Y. P. Liu and Z. B. Li, “The Homotopy Analysis Method for Approximating the Solution of the Modified Korteweg-de Vries Equation,” Chaos, Solitons and Fractals, Vol. 39, No. 1, 2009, pp. 1-8.

[13] Y. Bouremel, “Explicit Series Solution for the Glauert-Jet Problem by Means of the Homotopy Analysis Method,” Communication in Nonlinear Science Numerical Simulation, Vol. 12, No. 5, 2007, pp. 714-724.

[14] A. Molabahrami and F. Khani, “The Homotopy Analysis Method to Solve the Burgers-Huxley Equation,” Nonlinear Analysis Real World Applications, Vol. 10, No. 2, 2009, pp. 589-600.

[15] S. Abbasbandy, E. Magyari and E. Shivanian, “The Homotopy Analysis Method for Multiple Solutions of Nonlinear Boundary Value Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 9-10, 2009, pp. 3530-3536.

[16] H. N. Hassan and M. A. El-Tawil, “Solving Cubic and Coupled Nonlinear Schrodinger Equations Using the Homotopy Analysis Method,” International Journal of Applied Mathematics and Mechanics, Vol. 7, No. 8, 2011, pp. 41-64.

[17] H. N. Hassan and M. A. El-Tawil, “An Efficient Analytic Approach for Solving Two-Point Nonlinear Boundary Value Problems by Homotopy Analysis Method,” Mathematical Methods in the Applied Sciences, Vol. 34, No. 8, 2011, pp. 977-989.