Analog of the Darboux problem for a loaded integro-differential equation involving the Caputo fractional derivative

Abstract

In this paper, we prove the unique solvability of an analogue problem Darboux for a loaded integro-differential equation with Caputo operator by method of integral equations. The problem is equivalently reduced to a system of integral equations, which is unconditionally and uniquely solvable.

Cauchy problem for a high‐order loaded integro‐differential equation

“…It is to be shown that the statement is true for n initial value. Assume that the statement is true for any value of n = k. Then, it will be possible to prove that 34 the statement is true for n = k + 1. We actually break n = k + 1 into two parts; one part is n = k (which is already proved) and easily obtained to prove the other part.…”

“…Initial and boundary value problems for loaded second‐ and third‐order equations of hyperbolic, parabolic, elliptic, and mixed types were investigated in the works of previous research 20–34 . We would like to note that problems for the loaded equations of high‐order differential equations have not been studied yet.…”

“…Initial and boundary value problems for loaded second-and third-order equations of hyperbolic, parabolic, elliptic, and mixed types were investigated in the works of previous research. [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] We would like to note that problems for the loaded equations of high-order differential equations have not been studied yet. The construction of a theory of unique solvability in the study of the correctness of problem formulation for the generalized multidimensional differential equation with loaded term is required both for the internal completeness of the theory of loaded differential equation and for numerous applications.…”

Cauchy problem for a parabolic–hyperbolic equation with non‐characteristic line of type changing

“…Else, many researchers from different scientific fields are currently studying various types of advanced mathematical modelings using fractional operators, hybrid operators, and fractional differential equations with more general initial (Cauchy‐type problems) and boundary value conditions. Indeed, with these models, the processes can cover many applied problems and make it possible to study advanced fractional modelings and their generalized results 8–14 …”

“…Indeed, with these models, the processes can cover many applied problems and make it possible to study advanced fractional modelings and their generalized results. [8][9][10][11][12][13][14] Cauchy characteristic problem is seeking a solution of a partial differential equation (or a system of partial differential equations) which assumes prescribed values on a characteristic manifold. There is a large class of works, where problems of the Cauchy type are considered for the equation of parabolic, hyperbolic, and parabolic-hyperbolic types in the characteristic line of change of the type.…”