Initial and Initial-Boundary Value Problems for Multi-Parameter Integral-Differential Equations

This paper investigates initial and initial–boundary value problems for a class of multiparameter integral-differential equations. In particular, we formulate the Cauchy problem for an ordinary differential equation involving a multi-parameter integral-differential operator. The problem is equivalently transformed into a Volterra integral equation of the second kind. By applying the method of successive approximations, we derive an explicit representation of the solution. As an application of the newly established results, we study direct and inverse source problems for a partial differential equation that incorporates the same multi-parameter integral-differential operator in the time variable. An example is presented to illustrate the theoretical findings.

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