This current manuscript investigates a class of fractional order integro-differential equations in the context of conformable fractional derivative (CFD). The proposed class is analyzed for the existence of its solution, stability in the context of Ulam-Hyers (U-H), and also computed numerical solutions. Initially, we established conditions which are suffcient for the existence, uniqueness, and stability of solutions on the application of fixed point theorems and non-linear functional analysis. The U-H framework of stability ensures reliability, robustness which means the approximate solution must be nearly equal to the precise solution. The RK-4(Runge-Kutta) and Euler’s methods are used to address the numerical aspect. Furthermore, the simulations are also carried based on the solution evaluated through the mentioned numerical methods. On the basis of simulated results we have shown that the RK-4 method over the Euler’s method is more efficient and accurate. Due to the importance of artificial intelligence (AI) techniques, we use deep neural network (DNN) analysis to investigate a coupled system with fractional order derivative. To overcome this limitation, we recommend a DNN-based technique that calculate the approximate solution with free mesh manner, resulting computational load and maintaining accuracy. In particular, the application of fractional derivatives gives compatibility with DNN, increasing computational convergence and efficiency. We use Levenberg-Marquardt algorithm to examine a few probabilistic interpretations including mean squared error, root mean squared error and regression. An example from real world is provided at the end to strengthen our results of the study.
SHER et al. (Thu,) studied this question.