Exact solutions to linear and nonlinear wave and diffusion equations

  • Authors

    • Majeed AL-Jawary Head of Department of Mathematics, College of Education for Pure Sciences / Ibn-AL-Haithem, Baghdad University, Baghdad, Iraq
    2015-01-25
    https://doi.org/10.14419/ijamr.v4i1.4031
  • Diffusion equations, Exact solution, New iterative method, Wave equations.

  • Abstract

    In the present paper, the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] is used for solving linear and nonlinear wave and diffusion equations. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the

    difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations as in Homotopy Perturbation Method (HPM). A comparison with some existing techniques such as ADM, HPM and VIM also presented, which shows that the DJM is effective and convenient to use and overcomes the difficulties arising in existing techniques. The results show that the present method is very effective and simple and provide the analytic solutions. The software used for the calculations in this study was MATHEMATICA®8.0.

  • References

    1. [1] A.M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications. Beijing and Springer-Verlag Berlin Heidelberg, (2011).

      [2] A.M. Wazwaz, Exact solutions to nonlinear diffusion equations obtained by the decomposion method. Applied Mathematics and Computation 123 (2001) 109-122.

      [3] A.M.Wazwaz, The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations. Computers and Mathematics with Applications 54 (2007) 926-932.

      [4] A.M.Wazwaz,The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations. Computers and Mathematics with Applications 54 (2007) 933-939.

      [5] A. Sadighi, D.D. Ganji, Exact solutions of nonlinear diffusion equations by variational iteration method. Computers and Mathematics with Applications 54 (2007) 1112-1121.

      [6] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations. Journal of Mathematical Analysis and Applications 316 (2006) 753-763.

      [7] V. Daftardar-Gejji, H. Jafari, Solving a System of Nonlinear Functional Equations Using Revised New Iterative Method. World Academy of Science, Engineering and Technology 68 (2012) 08-21.

      [8] V. Daftardar-Gejji, H. Jafari, New iterative method: Application to partial differential equations. Applied Mathematics and Computation 203 (2008) 778-783.

      [9] V. Daftardar-Gejji, H. Jafari, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Computers and Mathematics with Applications 59 (2010) 1801-180.

      [10] M. Yaseen, M. Samraiz, S. Naheed, The DJ method for exact solutionsof Laplace equation. Results in Physics 3 (2013) 38-40.

      [11] V. Daftardar-Gejji, S. Bhalekar, An Iterative method for solving fractional differential equations. Proceedings in Applied Mathematics and Mechanics, 7(2007) 2050017-2050018.

      [12] S. Bhalekar, V. Daftardar-Gejji, Solving a System of Nonlinear Functional Equations Using Revised New Iterative Method. World Academy of Science, Engineering and Technology, 68(2012) 806-810.

      [13] M. Yaseen, M. Samraiz, The Modified New Iterative Method for Solving Linear and Nonlinear Klein-Gordon Equations. Applied Mathematical Sciences, 6(2012) 2979-2987.

      [14] M. Yaghouti, H. Deilami, Numerical Solution of Singular Differential-Difference Equations.World Applied Programming, 3(2013) 182-189.

      [15] I. Ullah, H. Khan and M. Rahim, Numerical Solutions of Higher Order Nonlinear Boundary Value Problems by New Iterative Method. Applied Mathematical Sciences, 7(2013) 2429-2439.

      [16] A. Hemeda, M. Al-luhaibi, New iterative method for solving gas dynamic equation. International Journal of Applied Mathematical Research, 3(2014) 190-195.

      [17] V. Daftardar-Gejji, H. Jafari, Convergence of the New Iterative Method. International Journal of Differential Equations, Article ID 989065, 10 pages.

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  • How to Cite

    AL-Jawary, M. (2015). Exact solutions to linear and nonlinear wave and diffusion equations. International Journal of Applied Mathematical Research, 4(1), 106-118. https://doi.org/10.14419/ijamr.v4i1.4031

    Received date: 2014-12-15

    Accepted date: 2015-01-12

    Published date: 2015-01-25