An optimization technique based on imperialist competition algorithm to measurement of error for solving initial and boundary value problems

•Converting a constrained optimization problem to unconstrained problem with penalty function.•Capability of imperialist competitive algorithm for solving and gaining better convergence.•Improving the computational efficiency and accuracy for solving initial and boundary value problems.•Present promising tool for solving higher-dimensional problems. Imperialist competitive algorithm (ICA) is proposed to solve initial and boundary value problems in this paper. A constrained problem is converted into an unconstrained problem through the use of a penalty method in other to define an appropriate fitness function that is optimized by means of the ICA method. The methodology adopted evaluates a large number of candidate solutions of the unconstrained problem with the ICA to minimize error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. The method is proficient approach to solve linear and nonlinear ODEs, systems of ordinary differential equations (SODEs), linear and nonlinear PDEs. Numerical experiments demonstrate the accuracy and efficiency of the proposed method. Thus, this method is a promising tool for solving higher-dimensional problems.