Abstract
In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.
Highlights
Fractional calculus became one of the intensively developing theories in modern mathematics due to its wide range of applications in real life processes and its generalized nature [1]
In recent papers [2, 3], fractional differential equations are used for modeling applications in blood alcohol and fish farm models and in [4] fractional partial differential equation is used for Frankl-Type Problem
Using special transformation (see (15)) we first reduced the considered Fractional order partial differential equations (FPDE) to the fractional ODEs and we solved these ODEs using successive iterative method
Summary
Fractional calculus became one of the intensively developing theories in modern mathematics due to its wide range of applications in real life processes and its generalized nature [1]. They considered the following fourth order degenerate PDE: xnut − tkuxxxx = 0, n, k = const > 0 They have got the equation with respect to ω: x3ωxxxx + (3 + c1 + c2 + c3)x2ωxxx +(1 + c1 + c2 + c3 + c1c2 + c1c3 + c2c3)xωxx+ (c1c2c3 − x)ωx − aω = 0, which has special solutions represented with hypergeometric functions pFq. The main motivation of the present research is the consideration of combinations of special fractional derivatives such as hyper-Bessel, Erdelyi-Kober. D0αt,δu(t, x) = D0βx,δu(t, x), 0 < α ≤ 1, 1 < β ≤ 2, and fractional differential equation involving hyper-Bessel operator in time and Erdelyi-Kober fractional derivative in space variable tθ. Definition 4. ( [25]) The left and right-sided Erdelyi-Kober fractional derivatives of order α, respectively, are given by (n − 1 < α < n, n ∈ N)
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