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Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

Erkinjon Karimov, Doniyor Usmonov, Maftuna Mirzaeva

Abstract

In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the superposition method. An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. Then, it is proved that the obtained solution indeed constitutes the solution of the considered problem.

Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

Abstract

In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the superposition method. An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. Then, it is proved that the obtained solution indeed constitutes the solution of the considered problem.
Paper Structure (4 sections, 2 theorems, 157 equations, 2 figures)

This paper contains 4 sections, 2 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Examples and numerical illustrations
  4. Conclusion

Key Result

Theorem 2.1

Let ${{\varphi }_{0}}\left( t \right),$${{\varphi }_{1}}\left( t \right)\in C\left[ 0;T \right],$$\tau \left( x \right)\in C\left[ 0;a \right],$${{t}^{1-\beta }}f\left( t,x \right)\in C\left( \overline{\Omega} \right)$ and $f\left( t,x \right)$ also satisfies the Hölder condition with respect to $x Then there exists a unique regular solution of the equation eq2.1 in the domain $\Omega,$ satisfyin

Figures (2)

  • Figure 1: Influence of the initial data.
  • Figure 2: Influence of the external force.

Theorems & Definitions (3)

  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.1