Numerical Solution of a Free Boundary Problem from Heat Transfer by the Second Kind Chebyshev Wavelets

Document Type: Original Paper

Author

Department of Mathematics, Faculty of science, Qom University of Technology, Qom, Iran

Abstract

In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra  integral equation of the first kind. Since the first kind integral equation is ill posed, and an appropriate method for such ill posed problems is based on wavelets, then we apply the Chebyshev wavelets to solve the integral equation. Numerical implementation of the method is illustrated by two benchmark problems originated from heat transfer. The behavior of the initial and free boundary heat functions along the position axis during the time have been shown through some three dimensional plots. The convergence of the method is pointed in the end of section 2. The numerical examples show the accuracy and applicability of the method from application and programming points of views.

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  1. Babayar-Razlighi B. and Ivaz K. Numerical solution of an inhomogeneous heat equation by the product integration method. The 43rd Annual Iranian Mathematics Conference: 515-518  (2012).
  2. Babayar-Razlighi B. and Solaimani M. Product integration method for numerical solution of a heat conduction problem. The 46rd Annual Iranian Mathematics Conference: 861-864 (2015).
  3. Orsi A. P. Product integration for Volterra integral equations of the second kind with weakly singular kernels. Math. Comp. 65(215): 1201-1212 (1996) .
  4. Criscuolo G.,  Mastroianni G. and Monegato G. Convergence properties of a class of product formulas for weakly singular integral equations. Math. Comp.  55(191): 213-230 (1990).
  5. Babayar-Razlighi B. and Soltanalizadeh B. Numerical solution for system of singular nonlinear volterra integro-differential equations by newton-product method. Appl. Math. Comput. 219: 8375-8383 (2013).
  6. Babayar-Razlighi B. and Soltanalizadeh B. Numerical solution of nonlinear singular volterra integral system by the newton-product integration method. Math. Comput. Model. 58: 1696-1703 (2013).
  7. Babayar-Razlighi B., Ivaz K. and Mokhtarzadeh M. Convergence of product integration method applied for numerical solution of linear weakly singular volterra systems.  B. Iran. Math. Soc. 37(3): 135-148 (2011).
  8. Babayar-Razlighi B., Ivaz K., Mokhtarzadeh M. and Badamchizadeh A., Newton-product integration for a two- phase stefan problem with kinetics.  B. Iran. Math. Soc. 38(4): 853-868 (2012).
  9. Babayar-Razlighi B., Ivaz K. and Mokhtarzadeh M. Newton-product integration for a stefan problem with kinetics. J. Sci. I. R. Iran. 22(1): 51-61 (2011).
  10. Abd-Elhameed W. M., Doha E. H. and Youssri Y. H. New Spectral second kind Chebyshev Wavelets Algorithm for Solving Linear and nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equation. Abstr. Appl. Anal. 2013: 9. Article ID 715756 (2013).
  11.  Biazar J. and Ebrahimi H. Chebyshev wavelets approach for nonlinear systems of volterra integral equations. Comput. Math. Appl. 63: 608–616 (2012)
  12. Cannon J. R. The one-dimensional heat equation. Addison-Wesley, Menlo Park, (1984).
  13. Boggess A. and Narcowich F. J. A first course in Wavelets with Fourier Analysis. Prentice Hall, (2001).
  14. Babolian E. and Shahsavaran A. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets. J. Comput. Appl. Math. 225: 87–95 (2009).
  15.  

    Restrepo J.M. and Leaf G.K. Inner product computations using periodized Daubechies wavelets, Int. J. Num. Methods Eng. 40: 3557–3578 (1997).
  1. S. A. Yousefi, Numerical solution of Abel-s integral equation by using legendre wavelets, Appl. Math. Comput. 175(1): 574-580 (2006).
  2.  Ghasemi M. and Tavassoli Kajani M. Numerical solution of time-varying delay systems by chebyshev wavelets, Appl. Math. Model. 35: 5235–5244 (2011).
  3.  Zhou F., Xu X., Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets. Appl. Math. Comput. 247: 353–367 (2014).
  4. Rudin W., Principles of Mathematical Analysis. McGraw Hill, New York (1964).