Fermat collocation method for fractional order variable coefficients differential equations and the system of such equations and residual error analysis

As it is known, due to the coefficients and structure of differential equations, different difficulties may arise at the point of analytical analysis. These difficulties are generally encountered with situations such as integrating in the domain of definition or non-linearity due to the structural features of the equations. In the method we discussed, we tried to find the coefficients of the Fermat polynomial solutions in the proposed series form, according to the algebraic equations to be formed with both fractional derivative and integral values in the fractional differential equation at points with certain properties in the domain of definition. Then, for the solution of the special problem where the initial conditions, boundary conditions, initial-boundary conditions or mixed conditions of the fractional differential equation were given, this information was processed into the system of equations found in the previous step, thus a final form for the system of algebraic equations was obtained. If there is a solution to this system of equations, this solution has given the necessary coefficients to write the solution of the fractional differential equation that meets the desired conditions. By substituting the obtained coefficients in the proposed solution polynomial, a solution in the semi-analytical Fermat polynomial structure was obtained. In case of exact solutions of the fractional differential equation, we determined how close this polynomial obtained to the exact solution and the maximum error in the approximation, besides, more precise numerical solutions were sought by adding the residual function to the solution polynomial in order to optimize and minimize the error. The steps we mentioned were theoretically first written step by step, and then noted as an algorithm we wrote, and the results obtained from the method we proposed were found quickly by using Maple program that worked at the optimum level, and again written by ourselves.