Solution of Nonlinear Random Partial Differential Equations by Using Finite Element Method

Authors

  • Ibrahim Elkott Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura
  • Ahmed Elsaid Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura
  • I. L. El-Kalla Department of Engineering Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura

DOI:

https://doi.org/10.26713/jims.v12i3.1405

Keywords:

Random differential equations, Random finite element method, Nonlinear partial differential equations

Abstract

In this paper, a new technique is proposed to solve some classes of nonlinear random partial differential equations using finite element method. Through this technique we were able to deal with the random variable in the presence of a nonlinear function. The idea of this technique is based on assuming that the nodal coefficients are functions of the random variable. Then by discretization of the random variable and using fitting over the discretized values of the random variable, and utilizing the shape functions of the finite element method, we get the approximate solution as a function in both space and random variable. Some numerical examples, in different domains, are presented to show the effectiveness of this technique.

Downloads

Download data is not yet available.

References

U. M. Ascher, R. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics (1995), DOI: 10.1137/1.9781611971231.

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer (2013), DOI: 10.1007/978-3-642-36519-5.

C. Burgos, J. C. Cortés, L. Villafuerte and R. J. Villanueva, Mean square calculus and random linear fractional differential equations, Applied Mathematics and Nonlinear Sciences 2(2) (2017), 317 – 328, DOI: 10.21042/AMNS.2017.2.00026.

J.-C. Cortés, L. Jódar, F. Camacho and L. Villafuerte, Random Airy type differential equations: Mean square exact and numerical solutions, Computers Mathematics with Applications 60(5) (2010), 1237 – 1244, DOI: 10.1016/j.camwa.2010.05.046.

A. M. El-Tawil and M. A. Sohaly, Mean square convergent three points finite difference scheme for random partial differential equations, Journal of the Egyptian Mathematical Society 20(3) (2012), 188 – 204, DOI: 10.1016/j.joems.2012.08.017.

D. V. Griffiths and G. A. Fenton, Probabilistic settlement analysis by stochastic and random finite element methods, Journal of Geotechnical and Geoenvironmental Engineering 135(11) (2009), 1629 – 1637, URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%29GT.1943-5606.0000126.

D. J. Higham, An algorithmic introduction to numerical simulation of Stochastic differential equations, SIAM Review 43(3) (2001), 525 – 546, URL: https://epubs.siam.org/doi/pdf/10.1137/S0036144500378302.

N. J. Higham and H. M. Kim, Numerical analysis of a quadratic matrix equation, IMA Journal of Numerical Analysis 20(4) (2000), 499 – 519, DOI: 10.1093/imanum/20.4.499.

D. V. Hutton, Fundamentals of Finite Element Analysis, McGraw-Hill (2004).

S. M. Iacus, Simulation and Inference for Stochastic Differential Equations: With R Examples, Springer Science Business Media (2009), URL: https://www.springer.com/gp/book/9780387758381.

S. L. Khalaf, Mean square solutions of second-order random differential equations by using homotopy perturbation method, International Mathematical Forum 6(48) (2011), 2361 – 2370, URL: http://m-hikari.com/imf-2011/45-48-2011/khalafIMF45-48-2011.pdf.

A. R. Khudair, A. A. Ameen and S. L. Khalaf, Mean square solutions of second-order random differential equations by using Adomian decomposition method, Applied Mathematical Sciences 5(51) (2011), 2521 – 2535, URL: http://www.m-hikari.com/ams/ams-2011/ams-49-52-2011/khalafAMS49-52-2011-2.pdf.

A. R. Khudair, A. A. Ameen and S. L. Khalaf, Mean square solutions of second-order random differential equations by using variational iteration method, Applied Mathematical Sciences 5(51) (2011), 2505 – 2519, URL: http://www.m-hikari.com/ams/ams-2011/ams-49-52-2011/khalafAMS49-52-2011-1.pdf.

A. R. Khudair, S. A. M. Haddad and S. L. Khalaf, Mean square solutions of second-order random differential equations by using the differential transformation method, Open Journal of Applied Sciences 6(4) (2016), 287, DOI: 10.4236/ojapps.2016.64028.

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Science 23 (2013), DOI: 10.1007/978-3-662-12616-5.

Y. W. Kwon and H. Bang, The Finite Element Method Using MATLAB, CRC Press (2018).

E. Madenci and I. Guven, The Finite Element Method and Applications in Engineering Using ANSYS, Springer (2015), URL: https://dl.acm.org/doi/book/10.5555/2746451.

X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Elsevier (2007), URL: https://www.elsevier.com/books/stochastic-differential-equations-and-applications/mao/978-1-904275-34-3.

M. A. Noor, Some iterative methods for solving nonlinear equations using homotopy perturbation method, International Journal of Computer Mathematics 87(1) (2010), 141 – 149.

K. Nouri, Study on efficiency of the Adomian decomposition method for stochastic differential equations, International Journal of Nonlinear Analysis and Applications 8(1) (2017), 61 – 68, URL: https://ijnaa.semnan.ac.ir/article_484.html.

P. E. Protter, Stochastic differential equations, in Stochastic Integration and Differential Equations, Springer, Berlin ” Heidelberg, 249 – 361 (2005), URL: https://www.springer.com/gp/book/9783540003137#.

J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis: With Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics, Oxford Univeristy Press, Oxford (2014).

C. Remani, Numerical Methods for Solving Systems of Nonlinear Equations, Lakehead University Thunder Bay, Ontario, Canada (2013), URL: https://www.lakeheadu.ca/sites/default/files/uploads/77/docs/RemaniFinal.pdf.

S. Ross, A First Course in Probability, Pearson Prentice Hall (2006), URL: http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf.

M. M. Saleh, I. L. El-Kalla and M. M. Ehab, Stochastic finite element based on stochastic linearization for stochastic nonlinear ordinary differential equations with random coefficients, in Proceedings of the 5th Wseas International Conference on Non-Linear Analysis, Non-Linear Systems and Chaos, 104 – 109 (2006), URL: https://dl.acm.org/doi/abs/10.5555/1974665.1974686.

M. M. Saleh, I. L. El-Kalla and M. M. Ehab, Stochastic finite element for stochastic linear and nonlinear heat equation with random coefficients, WSEAS Transactions on Mathematics 5(12) (2006), 1255 – 1262, URL: https://www.researchgate.net/publication/289351159_Stochastic_finite_element_for_stochastic_linear_and_nonlinear_heat_equation_with_random_coefficients.

G. Stefanou, The stochastic finite element method: past, present and future, Computer Methods in Applied Mechanics and Engineering 198(9-12) (2009), 1031 – 1051, DOI: 10.1016/j.cma.2008.11.007.

R. Steyer and W. Nagel, Probability and Conditional Expectation: Fundamentals for the Empirical Sciences, John Wiley Sons (2017), URL: https://www.wiley.com/en-in/Probability+and+Conditional+Expectation%3A+Fundamentals+for+the+Empirical+Sciences-p-9781119243526.

E. Süli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press (2003), URL: http://newdoc.nccu.edu.tw/teasyllabus/111648701013/Numerical_Analysis.pdf.

L. Villafuerte and B. M. Chen-Charpentier, A random differential transform method theory and applications, Applied Mathematics Letters 25(10) (2012), 1490 – 1494, DOI: 10.1016/j.aml.2011.12.033.

P. Wriggers, Nonlinear Finite Element Methods, Springer Science (2008), DOI: 10.1007/978-3-540-71001-1.

C. Yuan and X. Mao, Convergence of the Euler Maruyama method for stochastic differential equations with Markovian switching, Mathematics and Computers in Simulation 64(2) (2004), 223 – 235, URL: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.143.789&rep=rep1&type=pdf.

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, Butterworth Heinemann (2000).

Downloads

Published

2020-09-30
CITATION

How to Cite

Elkott, I., Elsaid, A., & El-Kalla, I. L. (2020). Solution of Nonlinear Random Partial Differential Equations by Using Finite Element Method. Journal of Informatics and Mathematical Sciences, 12(3), 233–246. https://doi.org/10.26713/jims.v12i3.1405

Issue

Vol. 12 No. 3 (2020)

Section

Research Articles

License

Authors who publish with this journal agree to the following terms:
  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.