Series Form Solution to Two Dimensional Heat Equation of Fractional Order

Year 2019, Volume: 2 Issue: 4, 193 - 199, 25.12.2019

Atta Ullah , Kamal Shah , Rahmat Ali Khan

Abstract

In this article we develop series type solution to two dimensional wave equation involving external source term of fractional order. For the require result, we use iterative Laplace transform. The solution is computed in series form which is rapidly convergent to exact value. Some examples are given to illustrate the establish results.

Keywords

Two dimensional wave equation, Series solution, Iterative Laplace transform

References

• [1] H. Eltayeb, Hassan, and A. Kiliçman, A note on solutions of wave, Laplace's and heat equations with convolution terms by using a double Laplace transform, Applied Mathematics Letters 21(12) (2008) 1324- 1329.
• [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [3] T. Khan, K. Shah, R.A. Khan and A. Khan, Solution of fractional order heat equation via triple Laplace transform in 2 dimensions, Mathematical Methods in the Applied Sciences 41(2) (2018): 818-825.
• [4] A.A. Kilbas, H. Srivastava and J. Trujillo, Theory and application of fractional differential equations, North Holland Mathematics Studies, vol. 204, Elseveir, Amsterdam, 2006.
• [5] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
• [6] V. Lakshmikantham and S. Leela, Naguma-type uniqueness result for fractional differential equations, Non- linear Anal., 71 (2009) 2886--2889.
• [7] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• [8] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.
• [9] Yu Z. Povstenko, Fractional heat conduction equation and associated thermal stress, Journal of Thermal Stresses 28(1) (2004) 83-102.
• [10] H. Richard, Elementary applied partial differential equations, Englewood Cliffs, NJ: Prentice Hall, 1983. [11] F.J. Rizzo and D.J. Shippy, A method of solution for certain problems of transient heat conduction, AIAA Journal 8(11) (1970) 2004-2009.
• [12] Y.A. Rossikhin, and M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997) 15--67.
• [13] K. Shah, H. Khalil and R.A. Khan, Analytical solutions of fractional order diffusion equations by natural transform method, Iranian Journal of Science and Technology, Transactions A: Science 42(3) (2018) 1479- 1490.
• [14] G.Spiga and M. Spiga, Two-dimensional transient solutions for crossflow heat exchangers with neither gas mixed, (1987): 281-286.
• [15] Y. Zhang, Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang- Laplace transform, Thermal Science 18(2) (2014) 677-681.