Existence and Uniqueness of The Solution of The Exact Problem For The Integro-Differential Heat Distribution Equation
Abstract
The study of integro-differential equations plays a fundamental role in mathematical physics, particularly in the analysis of heat dissipation processes. The existence and uniqueness of solutions to such equations are crucial for ensuring the reliability of theoretical models. The Cauchy problem for the integro-differential heat dissipation equation can be reformulated into an equivalent Volterra integral equation. Traditional approaches employ fundamental solutions and functional series to establish solvability conditions. While various studies have explored heat conduction problems with memory effects, there remains a need for rigorous proofs ensuring the uniqueness of solutions in the space of Hölder functions. This study aims to establish a complete proof of the existence and uniqueness of the solution to the integro-differential heat dissipation equation by utilizing the method of successive approximations and integral inequalities. The research demonstrates that the functional series converges uniformly within the given domain, ensuring the existence of a solution. Furthermore, through the application of the Gronwall–Bellman inequality, it is shown that the solution is unique. The use of the Hölder function space in proving the uniqueness and existence of solutions offers a refined approach to analyzing heat dissipation equations, strengthening the theoretical foundations of inverse problem theory. The findings contribute to mathematical physics by providing a rigorous framework for modeling heat distribution processes and ensuring the stability of integro-differential equation-based models in applied sciences.
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