This paper presents an approximate solution of nonlinear fractional differential equations (FDEs) that exhibit an oscillatory behavior by using a metaheuristic technique. The solutions of the governing equations are approximated by using homotopy perturbation method (HPM) along with the fractional derivative in the Caputo sense. The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm. The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm (CSA). The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution. The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased. The critical analysis is also provided by the numerical simulation of two different test models. Discussion of key points has been determined by the tabulation of numerical values and graphs. Comparative study of the results with known numerical technique is also performed.
This endeavor proposes an effective implementation of a hybrid technique for computing the approximate solution of fractional order Helmholtz equation, with Dirichlet boundary conditions. The novel scheme is an amalgamation of the traditional finite difference method with the bat optimization algorithm (BOA). This nature‐inspired optimization technique simulates the echolocation behavior of foraging bats, which ascertain the surroundings through the echo of their emitted sound pulse. Systematically, the deliberated fractional order system is altered into an integer order partial differential equation by virtue of linearized expansion of Laplace transformation. Subsequently, the attained system is processed through the proposed innovative finite difference optimization technique (FDOT) the numerical discussions. Furthermore, some experiments are carried out in order to expound the effective execution and application of the technique. In addition, the convergence and accuracy of the scheme are also delineated numerically, via statistical inference. The significant outcomes of the analysis reveal the advantageousness of the proposed numerical scheme, which can efficiently handle the complexities of the fractional order 2‐D Helmholtz equation.
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