In this work, we present results on the embeddings of fractional Riemann-Liouville Sobolev spaces, using an important relationship between Riemann-Liouville Sobolev spaces and ordinary Sobolev spaces.This relationship allows us to prove compact embeddings after establishing continuous embeddings based on the continuity of the Riemann-Liouville fractional integral operators between Lebesgue spaces under certain conditions.We provide an example of a boundary problem where existence and uniqueness are addressed using two methods: the fixed point method and the Faedo-Galerkin method.Both methods require specific fractional type embeddings.

This article investigates a time-fractional space-nonlocal diffusion equation in a bounded domain. The fractional operators are defined rigorously, using the Caputo fractional derivative of order  and the Riemann-Liouville fractional integral of order  , where 0 <  <  1 . The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated and the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

This study focuses on constructing solutions for a specific type of second order fractional differential equation involving nonlocal boundary conditions.We additionally provide exemplifications that demonstrate the application of our results.Mathematics subject classification (2020)

The main objective of this paper is to prove the existence and uniqueness of mild solution for abstract differential equations by using the resolvent operators and fixed point theorem. Moreover, we studied some examples on partial differential equation with Caputo-Fabrizio derivative.

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This work focuses on the semi-analytical methods for obtaining the solutions of time fractional partial integro-differential equations.Adomian decomposition method (ADM) and homotopy perturbation method (HPM) are successfully applied.Further, the modified version of homotopy perturbation method is applied which is comparatively more accurate than the other two methods.These methods are shown to be efficient and converge rapidly to the exact solution.Graphs are plotted and tabular data are recorded which represents the accuracy of the proposed techniques.

The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution is represented by Fourier-Bessel series in the given domain. It is established the connection between the given data and the unique solvability of the problem.

In this paper, we present some results on existence and uniqueness for a class of boundary value problems for iterative fractional differential equations with generalized Caputo fractional derivative.For our proofs, we employ some suitable fixed point theorems.Finally, we provide an illustration for more clarity.

A quasi-boundary method is used to study an ill-posed, time-fractional diffusion equation involving the fractional Riemann-Liouville derivative.In particular, we consider an ill-posed problem for a family of well-posed problems, and prove, by means of eigenfunction expansions, that the solutions of the latter problems converge to the solutions associated with the former problem.The analysis presented includes providing conditions for the rate of the convergence.