Abstract
The main goal of this study is to find the solution of initial boundary value problem for the one-dimensional time and space-fractional diffusion equation which is a very intriguing topic for many researchers. With the aim of newly defined inner product, which is the main contribution of this study, the analytic solution of the boundary value problem is obtained. The time and space-fractional derivatives are defined in the Caputo sense which is more suitable than Riemann-Liouville sense. We apply the separation of variables method to reduce the problem to two separate fractional ODEs. The generalized solution is constructed/formed in the form of a Fourier series with respect to the eigenfunctions of a certain eigenvalue problem. In order to obtain the coefficients of the Fourier series for the solution, we define a new inner product which is the key point of study.
Highlights
Since PDE of fractional order contributes to modeling for the wide range of processes and systems, including past memories, in various scientific research areas, it has become very intriguing topic for many scientists
Main reason of this trend is that using fractional derivative is global in nature whereas the integer derivative is local in nature
The Caputo derivatives are more useful than RiemannLiouville derivatives since the analysis of the mathematical models involving Caputo derivatives gives closer results to the analysis of ones including integer derivatives
Summary
Since PDE of fractional order contributes to modeling for the wide range of processes and systems, including past memories, in various scientific research areas, it has become very intriguing topic for many scientists. Main reason of this trend is that using fractional derivative is global in nature whereas the integer derivative is local in nature. This property makes fractional DEs the best possible choice in modeling physical problems involving past memory and/or delay and attracts growing number of researchers. The Mittag–Leffler function takes the role of the exponential function which is used in the determination of solutions for ODEs with integer derivatives
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