Solution of Nonlinear Stochastic Langevin’s Equation Using WHEP, Pickard and HPM Methods

Affiliation(s)

Department of Mathematics, Faculty of Science & Mansoura University, Mansoura, Egypt.

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

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

Department of Mathematics, Faculty of Science & Mansoura University, Mansoura, Egypt.

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

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

ABSTRACT

This paper introduces analytical and numerical solutions of the nonlinear Langevin’s equation under square nonlinearity with stochastic non-homogeneity. The solution is obtained by using the Wiener-Hermite expansion with perturbation (WHEP) technique, and the results are compared with those of Picard iterations and the homotopy perturbation method (HPM). The WHEP technique is used to obtain up to fourth order approximation for different number of corrections. The mean and variance of the solution are obtained and compared among the different methods, and some parametric studies are done by using Matlab.

KEYWORDS

Nonlinear Stochastic D.E; Langevin’s Equation; WHEP Technique; Picard Approximation; HPM Technique

Nonlinear Stochastic D.E; Langevin’s Equation; WHEP Technique; Picard Approximation; HPM Technique

Cite this paper

M. Hamed, M. El-Twail, B. El-desouky and M. El-Beltagy, "Solution of Nonlinear Stochastic Langevin’s Equation Using WHEP, Pickard and HPM Methods,"*Applied Mathematics*, Vol. 5 No. 3, 2014, pp. 398-412. doi: 10.4236/am.2014.53041.

M. Hamed, M. El-Twail, B. El-desouky and M. El-Beltagy, "Solution of Nonlinear Stochastic Langevin’s Equation Using WHEP, Pickard and HPM Methods,"

References

[1] K. De Feriet, “Random Solutions of Partial Differential Equations,” Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, 1956, pp. 199-208.

[2] R. Bharucha, “A Survey on the Theory of Random Functions,” The Institute of Mathematical Sciences, Matscience Report 31, 1965.

[3] V. Lo Dato, “Stochastic Processes in Heat and Mass Transport,” Probabilistic Methods in Applied Mathematics, Vol. 3(A), 1973, pp. 183-212.

[4] B. A. Georges, “Random Generalized Solutions to the Heat Equations,” Journal of Mathematical Analysis and Applications, Vol. 60, No. 1, 1977, pp. 93-102. http://dx.doi.org/10.1016/0022-247X(77)90051-8

[5] W. T. Coffey and Y. P Kalmykov, “The Langevin Equation, with Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering,” 3rd Edition, World Scientific Publishing Company, Singapore City, 2012.

[6] M. A. El-Tawil, “The Application of the WHEP Technique on Partial Differential Equations,” International Journal of Differential Equations and Applications, Vol. 7, No. 3, 2003, pp. 325-337.

[7] M. A. El-Beltagy and M. A. El-Tawil, “Toward a Solution of a Class of Non-Linear Stochastic perturbed PDEs Using Automated WHEP Algorithm,” Applied Mathematical Modelling, 2013, in Press. http://dx.doi.org/10.1016/j.apm.2013.01.038

[8] M. El Tawil and W. Shawky “Wiener Functional, Integrals, and Stochastic Differential Equations Solutions Using EulerMaruyama, Picard and WHEP Computer Simulation Stud,” BSc Thesis, Engineering faculty, Cairo University, Cairo, 2011.

[9] M. A. El-Tawil and A. Fareed, “Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods,” Open Journal of Discrete Mathematics, Vol. 1, No. 1, 2011, pp. 6-21.

http://dx.doi.org/10.4236/ojdm.2011.11002

[10] M. El-Tawil and N. El-Molla, “The Approximate Solution of a Nonlinear Diffusion Equation Using Some Techniques, a Comparison Study,” Applied Mathematics and Computing, Vol. 29, No. 1-2, 2009, pp. 281-299.

[11] A. S. El-Johani, “Comparisons between WHEP and Homotopy Perturbation Techniques in Solving Stochastic Cubic Oscillatory Problems,” AIP Conference Proceedings, Vol. 1148, 2010, pp. 743-752. http://dx.doi.org/10.1063/1.3225426

[12] J. C. Cortes, J. V. Romero, M. D. Rosello and R. J. Villanueva, “Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain),” American Journal of Computational Mathematics, Vol. 2, No. 4, 2012, pp. 274-281. http://dx.doi.org/10.4236/ajcm.2012.24037

[13] N. Wiener, “Nonlinear Problems in Random Theory,” MIT Press, John Wiley, Cambridge, 1958.

[14] R. H. Cameron and W. T. Martin, “The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals,” Annals of Mathematics, Vol. 48, No. 2, 1947, pp. 385-392.

http://dx.doi.org/10.2307/1969178

[15] T. Imamura, W. Meecham and A. Siegel, “Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals,” Journal of Mathematical Physics, Vol. 6, No. 5, 1965, pp. 695-706.

http://dx.doi.org/10.1063/1.1704327

[16] W. C. Meecham and D. T. Jeng, “Use of the Wiener-Hermite Expansion for Nearly Normal Turbulence,” Journal of Fluid Mechanics, Vol. 32, No. 2, 1968, pp. 225-235.

http://dx.doi.org/10.1017/S0022112068000698

[17] R. Riganti and N. Bellomo, “Nonlinear Stochastic Systems in Physics and Mechanics,” World Scientific Publishing Co., Singapore City, 1987.

[18] M. El-Beltagy and A. Al-Johani, “Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation,” Engineering, Vol. 5, No. 5A, 2013, pp. 57-69.

[19] E. A. Ibijola and B. J. Adegboyegun, “A Comparison of Adomian’s Decomposition Method and Picard Iterations Method in Solving Nonlinear Differential Equations,” Nigeria Global Journal of Science Frontier Research Mathematics and Decision Sciences, Vol. 12, No. 7, 2012.

[20] I. K. Youssef, “Picard Iteration Algorithm Combined with Gauss-Seidel Technique for Initial Value Problems,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 345-355. http://dx.doi.org/10.1016/j.amc.2007.01.058

[21] R. Riganti and N. Bellomo, “Nonlinear Stochastic Systems in Physics and Mechanics,” World Scientific Publishing Co., Singapore City, 1987.

[22] J. H. He, “Application of Homotopy Perturbation Method to Nonlinear Wave Equations,” Chaos Solitons Fractals, Vol. 26, No. 3, 2005, pp. 295-300. http://dx.doi.org/10.1016/j.chaos.2005.03.006

[23] J. H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88.

[1] K. De Feriet, “Random Solutions of Partial Differential Equations,” Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, 1956, pp. 199-208.

[2] R. Bharucha, “A Survey on the Theory of Random Functions,” The Institute of Mathematical Sciences, Matscience Report 31, 1965.

[3] V. Lo Dato, “Stochastic Processes in Heat and Mass Transport,” Probabilistic Methods in Applied Mathematics, Vol. 3(A), 1973, pp. 183-212.

[4] B. A. Georges, “Random Generalized Solutions to the Heat Equations,” Journal of Mathematical Analysis and Applications, Vol. 60, No. 1, 1977, pp. 93-102. http://dx.doi.org/10.1016/0022-247X(77)90051-8

[5] W. T. Coffey and Y. P Kalmykov, “The Langevin Equation, with Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering,” 3rd Edition, World Scientific Publishing Company, Singapore City, 2012.

[6] M. A. El-Tawil, “The Application of the WHEP Technique on Partial Differential Equations,” International Journal of Differential Equations and Applications, Vol. 7, No. 3, 2003, pp. 325-337.

[7] M. A. El-Beltagy and M. A. El-Tawil, “Toward a Solution of a Class of Non-Linear Stochastic perturbed PDEs Using Automated WHEP Algorithm,” Applied Mathematical Modelling, 2013, in Press. http://dx.doi.org/10.1016/j.apm.2013.01.038

[8] M. El Tawil and W. Shawky “Wiener Functional, Integrals, and Stochastic Differential Equations Solutions Using EulerMaruyama, Picard and WHEP Computer Simulation Stud,” BSc Thesis, Engineering faculty, Cairo University, Cairo, 2011.

[9] M. A. El-Tawil and A. Fareed, “Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods,” Open Journal of Discrete Mathematics, Vol. 1, No. 1, 2011, pp. 6-21.

http://dx.doi.org/10.4236/ojdm.2011.11002

[10] M. El-Tawil and N. El-Molla, “The Approximate Solution of a Nonlinear Diffusion Equation Using Some Techniques, a Comparison Study,” Applied Mathematics and Computing, Vol. 29, No. 1-2, 2009, pp. 281-299.

[11] A. S. El-Johani, “Comparisons between WHEP and Homotopy Perturbation Techniques in Solving Stochastic Cubic Oscillatory Problems,” AIP Conference Proceedings, Vol. 1148, 2010, pp. 743-752. http://dx.doi.org/10.1063/1.3225426

[12] J. C. Cortes, J. V. Romero, M. D. Rosello and R. J. Villanueva, “Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain),” American Journal of Computational Mathematics, Vol. 2, No. 4, 2012, pp. 274-281. http://dx.doi.org/10.4236/ajcm.2012.24037

[13] N. Wiener, “Nonlinear Problems in Random Theory,” MIT Press, John Wiley, Cambridge, 1958.

[14] R. H. Cameron and W. T. Martin, “The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals,” Annals of Mathematics, Vol. 48, No. 2, 1947, pp. 385-392.

http://dx.doi.org/10.2307/1969178

[15] T. Imamura, W. Meecham and A. Siegel, “Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals,” Journal of Mathematical Physics, Vol. 6, No. 5, 1965, pp. 695-706.

http://dx.doi.org/10.1063/1.1704327

[16] W. C. Meecham and D. T. Jeng, “Use of the Wiener-Hermite Expansion for Nearly Normal Turbulence,” Journal of Fluid Mechanics, Vol. 32, No. 2, 1968, pp. 225-235.

http://dx.doi.org/10.1017/S0022112068000698

[17] R. Riganti and N. Bellomo, “Nonlinear Stochastic Systems in Physics and Mechanics,” World Scientific Publishing Co., Singapore City, 1987.

[18] M. El-Beltagy and A. Al-Johani, “Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation,” Engineering, Vol. 5, No. 5A, 2013, pp. 57-69.

[19] E. A. Ibijola and B. J. Adegboyegun, “A Comparison of Adomian’s Decomposition Method and Picard Iterations Method in Solving Nonlinear Differential Equations,” Nigeria Global Journal of Science Frontier Research Mathematics and Decision Sciences, Vol. 12, No. 7, 2012.

[20] I. K. Youssef, “Picard Iteration Algorithm Combined with Gauss-Seidel Technique for Initial Value Problems,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 345-355. http://dx.doi.org/10.1016/j.amc.2007.01.058

[21] R. Riganti and N. Bellomo, “Nonlinear Stochastic Systems in Physics and Mechanics,” World Scientific Publishing Co., Singapore City, 1987.

[22] J. H. He, “Application of Homotopy Perturbation Method to Nonlinear Wave Equations,” Chaos Solitons Fractals, Vol. 26, No. 3, 2005, pp. 295-300. http://dx.doi.org/10.1016/j.chaos.2005.03.006

[23] J. H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88.