On the challenge of identifying space dependent coefficient in space-time fractional diffusion equations by fractional scaling transformations method

Year 2022, Volume: 7 Issue: 2, 132 - 145, 30.09.2022

Mine Aylin Bayrak , Ali Demir

Abstract

In this study, we get over the challenge of recovering unknown space dependent coefficient in space-time fractional diffusion equations by means of fractional scaling transformations method. Fractional differential equation is given in the sense of the conformable fractional derivative having substantial properties. By these properties and fractional scaling transformations method the fractional problem is reduced into integer order problem which allows us to tackle the problem better. Then we establish the solution and unknown coefficient of the reduced problem. Later, by employing inverse transformation, the solution and unknown coefficient of the fractional problem are obtained. Finally, some examples are presented to illustrate the implementation and effectiveness of the method.

Keywords

space time fractional diffusion equation, fractional scaling transformation method, conformable fractional derivative

References

• 1. Oldham, K. B.and Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, (Academic Press,1974).
• 2.} Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, (John Wiley and Sons, 1993).
• 3. Debnath, L. A. Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003).
• 4.} Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations, (Elsevier, 2006).
• 5. Podlubny, I. Fractional differential equation, San Diego, CA: Academic Press, 1999.