An Extended Analytical and Numerical Study the Nonlocal Boundary Value Problem for the Functional Integro-Differential Equation with the Different Conditions

In this work, we present a study of the nonlocal functional integro-differential equation with the nonlocal conditions. This study is different from the rest of the previous studies as we do the analytical study by studying the existence and uniqueness of the solution and also the continuous dependence of the proposed system. In addition, we apply all results of the analytical studying to some examples and find the exact solutions for them using the modified decomposition method. Also, we offer a numerical study of this system, unless it has been previously studied for the method of solving the proposed examples numerically using the finite difference-Simpson’s method. Some comparisons of numerical solutions are given with exact solutions to show the accuracy of the methods used, in addition to some figures that illustrate this. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.