New Approach for Solving Partial Differential Equations Based on Collocation Method

Alaa Almosawi; Luma N. M. Tawfiq

doi:10.24297/jam.v18i.8733

Authors

Alaa Almosawi University of Al-Qadisiyah
Luma N. M. Tawfiq University of Baghdad

DOI:

https://doi.org/10.24297/jam.v18i.8733

Keywords:

Partial Differential Equations, Integral Transform, LA-Transform, Collocation Method

Abstract

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion.

Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

Downloads

Download data is not yet available.

Author Biographies

Alaa Almosawi, University of Al-Qadisiyah

Department of Mathematics, College of Education, Iraq

Luma N. M. Tawfiq, University of Baghdad

Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham,

References

Abbasbandy, S. and Shivanian, E. (2009). Solution of Singular Linear Vibrational BVPs by the Homotopy Analysis Method. Journal of Numerical Mathematics and Stochastics, 1(1), 77-84. URL: http://www.jnmas.org/volume-1.html.

Liao, S. (2009). Notes on the homotopy analysis method: Some definitions and theorems. Communications in Nonlinear Science and Numerical Simulation, 14(4), 983–997. doi: 10.1016/j.cnsns.2008.04.013

Mastroberardino, A. (2001). Homotopy analysis method applied to electrohydrodynamic flow. Communications in Nonlinear Science and Numerical Simulation, 16, 2730-2736. doi: 10.1016/j.cnsns.2010.10.004.

Gupta, V.G. and Gupta, S. (2012). Application Of Homotopy Analysis Method For Solving Nonlinear Cauchy Problem. Surveys in Mathematics and its Applications, 7, 105-116. URL: http://www.utgjiu.ro/math/sma/v07/a08.html.

Kurulay, M. and Secer, A., (2013). A New Approximate Analytical Solution of Kuramoto - Sivashinsky Equation Using Homotopy Analysis Method. Applied Mathematics & Information Sciences, 7(1), 267-271. doi: 10.12785/amis/070133.

Enadi, M. O., and Tawfiq, L.N.M. (2019). New Technique for Solving Autonomous Equations. Ibn Al-Haitham Journal for Pure and Applied Science, 32(2), 123-130. doi: 10.30526/32.2.2150.

He, J.-H. (2006). Homotopy perturbation method for solving boundary value problems. Physics Letters A, vol. 350(1-2), 87-88. doi: 10.1016/j.physleta.2005.10.005.

Jafari, H. and Saeidy, M. (2008). Application of Homotopy Perturbation Method for Solving Gas Dynamics Equation. Applied Mathematical Sciences, 2(48), 2393-2396. URL: http://www.m-hikari.com/ams/ams-password-2008/ams-password45-48-2008/index.html.

Yu, J., and Huang, J.-G. (2010). Application of Homotopy Perturbation Method for the Reaction-diffusion Equation. International Journal of Nonlinear Sciences and Numerical Simulation, 11(Supplement). doi: 10.1515/ijnsns.2010.11.s1.61.

Afrouzi, G., Ganji, D. D., Hosseinzadeh, H., and Talarposhti, R. (2011). Fourth Order Volterra Integro-differential Equations Using Modifed Homotopy-perturbation Method. Journal of Mathematics and Computer Science, 3(2), 179-191. doi: 10.22436/jmcs.03.02.10.

Desai, K. R. and Pradhan,V. H. (2013). Solution by Homotopy Perturbation Method of Linear and Nonlinear Diffusion Equation. International Journal of Emerging Technology and Advanced Engineering, 3(4) URL: https://www.ijetae.com/Volume3Issue4.html.

Rach, R. (1987). On the Adomian (decomposition) method and comparisons with Picards method. Journal of Mathematical Analysis and Applications, 128(2), 480-483. doi: 10.1016/0022-247x(87)90199-5.

Hosseini, M., and Nasabzadeh, H. (2007). Modified Adomian decomposition method for specific second order ordinary differential equations. Applied Mathematics and Computation, 186(1), 117-123. doi: 10.1016/j.amc.2006.07.094.

Abassy, T. A. (2010). Improved Adomian decomposition method. Computers and Mathematics with Applications, 59(1), 42-54. doi: 10.1016/j.camwa.2009.06.009.

Al-Hayani, W. (2011). Adomian decomposition method with Green's function for sixth-order boundary value problems. Computers and Mathematics with Applications, 61(6), 1567-1575. doi: 10.1016/j.camwa.2011.01.025.

Al-Hayani, W. (2014). Adomian Decomposition Method with Green's Function for Solving Tenth-Order Boundary Value Problems. Applied Mathematics, 5(10), 1437-1447. doi:10.4236/am.2014.510136.

Agom, E. U., and Ogunfiditimi, F. O. (2016). Numerical Application of Adomian Decomposition Method to One Dimensional Wave Equations. International Journal of Science and Research (IJSR), 5(5), 2306-2309. doi: 10.21275/v5i5.nov162419.

Momani, S., Odibat, Z., and Alawneh, A. (2007). Variational iteration method for solving the space- and time-fractional KdV equation. Numerical Methods for Partial Differential Equations, 24(1), 262-271. doi: 10.1002/num.20247.

Salehpoor, E., and Jafari, H. (2011). Variational Iteration Method A Tools For Solving Partial Differential Equations. Journal of Mathematics and Computer Science, 2(2), 388-393. doi: 10.22436/jmcs.002.02.18.

Wang, Q., and Fu, F. (2012). Variational Iteration Method for Solving Differential Equations with Piecewise Constant Arguments. International Journal of Engineering and Manufacturing, 2(2), 36-43. doi: 10.5815/ijem.2012.02.06.

Tawfiq, L.N.M.; Naoum, R.S. (2007). Density and approximation by using feed forward Artificial neural networks. Ibn Al-Haitham Journal for Pure and Applied Sciences, 20(1), 67-81. URL: http://jih.uobaghdad.edu.iq/index.php/j/article/view/1335.

Effati, S., and Pakdaman, M. (2010). Artificial neural network approach for solving fuzzy differential equations. Information Sciences, 180(8), 1434-1457. doi: 10.1016/j.ins.2009.12.016.

Tawfiq, L. N. M.; Oraibi, Y. A. (2013). Fast Training Algorithms for Feed Forward Neural Networks. Ibn Al-Haitham Journal for Pure and Applied Science, 26(1), 275-280. :URL: http://jih.uobaghdad.edu.iq/index.php/j/article/view/534.

Ali MH, Tawfiq LNM, Thirthar A. A. (2019). Designing Coupled Feed Forward Neural Network to Solve Fourth Order Singular Boundary Value Problem. Revista Aus, 26(2), 140-146. doi: 10.4206/aus.2019.n26.2.20. URL: http://www.ausrevista.com/26-2.html.

Tawfiq, L.N.M. and Salih, O. M. (2019). Design neural network based upon decomposition approach for solving reaction diffusion equation. Journal of Physics: Conference Series, 1234(1234 012104), 1-8. doi: 10.1088/1742-6596/1234/1/012104.

Az-Zo'bi, E. (2012). Modified Laplace Decomposition Method. World Applied Sciences Journal, 18(11), 1481-1486. URL: https://www.idosi.org/wasj/wasj18(11)/1.pdf. doi: 10.5829/idosi.wasj.2012.18.11.1522.

Osama H. Mohammed, Huda A. Salim. (2018). Computational methods based laplace decomposition for solving nonlinear system of fractional order differential equations. Alexandria Engineering Journal, 57(4), 3549-3557. doi: 10.1016/j.aej.2017.11.020.

Eltayeb, H., & Kılıçman, A. (2012). Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations. Abstract and Applied Analysis, pp. 1-13. doi: 10.1155/2012/412948.

Ramadan, M. A. And Al-Luhaibi, M. S. (2014). Application of Sumudu Decomposition Method for Solving Linear and Nonlinear Klein-Gordon Equations. International Journal of Soft Computing and Engineering (IJSCE), 3(6), 138-141. URL: https://www.ijsce.org/download/volume-3-issue-6/.

D. Ziane, D. Baleanu, K. Belghaba, M. Hamdi Cherif. (2019). Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative. Journal of King Saud University - Science, 31(1), 83-88. doi: 10.1016/j.jksus.2017.05.002.

Tawfiq, L.N.M. (2016). Using collocation neural network to solve Eigenvalue problems. MJ Journal on Applied Mathematics, 1(1), 1-8. doi:10.1155/2014/906376.

Salih H, Tawfiq LNM, Yahya ZRI, Zin S M. (2018). Solving Modified Regularized Long Wave Equation Using Collocation Method. Journal of Physics: Conference Series, 1003(012062), 1-10. doi :10.1088/1742-6596/1003/1/012062.

Enadi MO, Tawfiq LNM. (2019). New Approach for Solving Three Dimensional Space Partial Differential Equation. Baghdad Science Journal, 16(3), 786-792. doi:10.21123/bsj.2019.16.3(Suppl.).0786. URL: http:// bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/4153.

Jabber, A. K. and Tawfiq, L. N. M. (2018). New Transform Fundamental Properties and Its Applications. Ibn Al-Haitham Jour. for Pure and Appl. Sci., 31(2), 151-163. doi: 10.30526/31.2.1954. URL: http://jih.uobaghdad.edu.iq/index.php/j/article/view/1954.