Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method

In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order μ ∈ 0,1 with λ ∈ ℤ + , θ , σ ∈ ℝ + is presented as follows: D t μ C u x , t + θ u λ x , t u x x , t − σ u x x t x , t = 0 , where for λ = 1 , θ = 1 , σ = 1 s and λ = 2 , θ = 3 , σ = 1 , equations are changed into the equal width and modified equal width equations, respectively. The analytical method which we have used for solving this equation is based on a combination of the homotopy perturbation method and Sadik transform. The convergence and error analysis are discussed in this article. Plots of the analytical results with three examples are presented to show the applicability of this numerical method. Comparison between the obtained absolute errors by the suggested method and other methods is demonstrated.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ε-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηTpoly(log(nT/ε))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ε.

Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.