Solution of differential equations using monte carlo methods

In this thesis, the effectiveness of Monte Carlo methods in the numerical solution of differential equations was investigated, and the applicability of these methods to various types of differential equations was examined in detail. Specifically, approximate solutions for singular, singularly perturbed, delayed, and Volterra-type integro-differential equations were addressed. The core approach of the study involves transforming differential equations into integral equations and solving these new formulations using Monte Carlo methods. Some examples in the thesis were also solved using other integral methods, and comparisons were made. Monte Carlo methods are statistical approaches that enable the derivation of approximate solutions by performing random sampling in the solution space. In this study, the flexibility and powerful approximation capabilities of Monte Carlo methods were emphasized in cases where classical deterministic numerical methods fall short or become challenging to apply. By using random numbers, approximate values of integral expressions were calculated, thereby obtaining solutions to the relevant differential equations. Within the scope of the thesis, the theoretical foundations of the Monte Carlo method were first presented, and random number generation and sampling techniques were detailed. Subsequently, integral formulations for different types of differential equations were derived, and their solutions were obtained using the Monte Carlo approach. The applications performed on integral equations demonstrate the broad-ranging potential of the Monte Carlo method.