Improving the analytical method of convergence using Runge-Kutta optimization algorithm for solving fractional-order differential equations

Solving linear differential equations of fractional ranks is shown the steps for solving them using the homotopy analytical method with the Runge-Kutta algorithm for their optimization. We took numerical examples and solved them approximately with the sequence that we got from collecting the iterations of the homotopy method and then taking the final sequence and finding the average square of the error and considering it as a fitness function in the Runge-Kutta algorithm for optimization and finding the best value of the auxiliary parameter h in the virtual homotopy method and taking the fewest Steps for solving linear differential equations with fractional ranks using (HAM-OBE) This algorithm is a process of optimization of the results of the homotopy analytical method using a method (rang-Kota optimization algorithm) the method can be summarized in five steps. 1. We take a Linear Differential Equation of fractional rank and calculate the initial guess using the equation y0(x)=∑k=0n−1y(k)(0+)xkk!.2.⁠. Finding subsequent iterations using the equation ym(x)=xmym−1(x)+hIa(Dxaym−1(x)+cym−1(x)) Using the first iteration as a basis for finding a solution to a differential equation with fractional ranks by the homotopy analytical method (HAM) to arrive at the solution series according to specific iterations. 3. The last iteration of the last 𝑄4 is taken, which is the product of summing the resulting iterations 𝑄4=𝑦0+𝑦1+𝑦2+𝑦3(we will take four iterations in this search). 4. The Runge-Kutta optimization algorithm (HAM-OBE) is used to find the best value of the parameter (h) to improve the results of the homotopy analytical method for solving differential equations with fractional ranks. We used the mean squared error as an appropriate function in the algorithm and a criterion to determine the improvement of results, such as the exact solution. 5. The final results of the two methods (HAM-OBE) and homotopy (HAM) are compared with the exact solution using the error ratio (RMSE). In this research, we solve fractional-order initial value problems of the Caputo type using the homotopy Analysis Method (HAM) with the Runge-Kutta Optimization Algorithm(HAM-OBE). The (HAM-OBE) selects the best value for the parameter (h), thereby improving the HAM's results. We have proven this by taking three examples and comparing the mean squared error results of the (HAM) with the(HAM-OBE). It has also been shown that the (HAM-OBE) method is reliable, efficient, and better than previous methods through plotting using MATLAB software.