Application of crank-nicolson method to a random component heat equation

In this study, the solution of a random component heat equation is obtained by using Crank-Nicolson Method. The initial condition of this equation is examined by Normal distribution. The expected value and variance of solution of this equation are obtained. Crank-Nicolson method is applied to analyze the solution of this equation. Also, the solution and the graphs of the expected value and variance are obtained by using MATLAB software. The results of the heat equation are compared with random characteristics of this equation. Firstly, a random component heat equation is solved by this method.

___

  • [1] Ayaz, F. (2004). Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 147(2), 547-567.
  • [2] Babuška, I., Nobile, F., & Tempone, R. (2007). A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis, 45(3), 1005-1034.
  • [3] Charrier, J. (2012). Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM Journal on numerical analysis, 50(1), 216-246.
  • [4] Crank, J., & Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc., 43(1), 50-67, doi:10.1007/BF02127704.
  • [5] Cortés, J.C., Jódar, L., Villafuerte, L., & Villanueva, R.J. (2007). Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3), 44-48.
  • [6] Çelik, C., & Duman, M. (2012). Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, Journal of computational physics, 231(4), 1743-1750.
  • [7] Ekolin, G. (1991). Finite difference methods for a nonlocal boundary value problem for the heat equation. BIT 31, 245-261.
  • [8] El-Tawil, & Sohaly, M.A. (2012). Mean square convergent three points finite difference scheme for random partial differential equations. J. of the Egyptian Math. Soc., 20(3), 188-204.
  • [9] Feller, W. (2008). An introduction to probability theory and its applications (Vol. 2). John Wiley & Sons.
  • [10] Gunzburger, M. D., Webster, C. G., & Zhang, G. (2014). Stochastic finite element methods for partial differential equations with random input data. Acta Numerica, 23, 521-650.
  • [11] He, J.H. (1999). Variational iteration method-a kind of non-linear analytical technique:some examples. Int. J. of Non-Linear Mech., 34(4), 699-708.
  • [12] He, J.H. (2003). Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and computation, 135(1), 73-79.
  • [13] He, J.H. (2006). Homotopy perturbation method for solving boundary value problems. Physics letters A, 350(1-2), 87-88.
  • [14] He, J.H. (2006). Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 20(18), 2561-2568.
  • [15] Kangalgil, F., & Ayaz, F. (2009). Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals, 41(1), 464-472.
  • [16] Khudair, A.R., Ameen, A.A., & Khalaf, S.L. (2011). Mean square solutions of second-order random differential equations by using adomian decomposition method. Applied Mathematical Sciences, 5, 2521-2535.
  • [17] Kuo, F. Y., Schwab, C., & Sloan, I. H. (2012). Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM Journal on Numerical Analysis, 50(6), 3351-3374.
  • [18] Mathews, J.H., & Fink, K.D. (2004). Numerical methods using MATLAB (Vol. 4). Upper Saddle River, NJ: Pearson Prentice Hall.
  • [19] Merdan, M. (2010). A new application of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk Journal of Applied Mathematics, 11(2), 27-40. [20] Merdan, M., Anac, H., Bekiryazici, Z., & Kesemen, T. (2019). Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Pade Method. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(1), 108-118.
  • [21] Nobile, F., Tempone, R., & Webster, C. G. (2008). An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 46(5), 2411-2442.
  • [22] Odibat, Z., & Momani, S. (2008). Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos, Solitons & Fractals, 36(1), 167-174.
  • [23] Smith, G.D. (1965). Numerical Solution of Partial Differential Equations. Oxford University Press.
  • [24] Thirumalai, S., Seshadri, R., & Yuzbasi, S. (2019). Population dynamics between a prey and a predator using spectral collocation method. International Journal of Biomathematics, 1950049.
  • [25] Yüzbaşı, Ş. (2017). A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions. International Journal of Computational Methods, 14(02), 1750015.
  • [26] Yüzbaşi, Ş., & Ismailov, N. (2017). Differential Transform Method to Solve Two-Dimensional Volterra Integral Equations with Proportional Delays. New Trends in Mathematical Sciences, 5(4), 65-71.
  • [27] Yüzbaşi, Ş., & Karaçayir, M. (2017). Application of homotopy perturbation method to solve two models of delay differential equation systems. International Journal of Biomathematics, 10(06), 1750080.
  • [28] Yüzbaşı, Ş. (2018). A collocation approach for solving two-dimensional second-order linear hyperbolic equations. Applied Mathematics and Computation, 338, 101-114.
  • [29] Yüzbaşı, Ş., & Karaçayır, M. (2018). A Galerkin-Type Method to Solve One-Dimensional Telegraph Equation Using Collocation Points in Initial and Boundary Conditions. International Journal of Computational Methods, 15(05), 1850031.
  • [30] Zhou, J.K. (1986). Differential Transform and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan, China.