Homotopy Perturbation Transform Method Research Articles

Sumudu transform HPM for Klein-Gordon and Sine-Gordon equations in one dimension from an analytical aspect

In the present research work, a hybrid algorithm is introduced, which includes an integral transform “Sumudu Transform” and the well-known semi-analytical regime “Homotopy Perturbation Method” named as “Sumudu Transform Homotopy Perturbation Method (STHPM)” to evaluate the exact solution of Klein-Gordon and Sine-Gordon equations. The discussed equations in this research have a prominent role in sciences and engineering. The authenticity and efficacy of this regime are established via a comparison between approximated solutions and exact solutions. Convergence analysis is also provided, which affirms that the solution obtained from STHPM is convergent and unique in nature. The results obtained by STHPM are compared with exact solutions. 2D and 3D plots are also discussed. The present regime is a reliable technique to provide the exact solution to a wide category of non-linear PDEs in an easy way, without any need of discretization, complex computation, linearization, and it is also error-free.

A Semianalytical Approach for the Solution of Nonlinear Modified Camassa–Holm Equation with Fractional Order

This paper presents the approximate solution of the nonlinear acoustic wave propagation model is known as the modified Camassa–Holm (mCH) equation with the Caputo fractional derivative. We examine this study utilizing the Laplace transform (ℒ T) coupled with the homotopy perturbation method (HPM) to construct the strategy of the Laplace transform homotopy perturbation method (ℒ T‐HPM). Since the Laplace transform is suitable only for a linear differential equation, therefore ℒ T‐HPM is the suitable approach to decompose the nonlinear problems. This scheme produces an iterative formula for finding the approximate solution of illustrated problems that leads to a convergent series without any small perturbation and restriction. Graphical results demonstrate that ℒ T‐HPM is simple, straightforward, and suitable for other nonlinear problems of fractional order in science and engineering.

A Few Methods of Solving Optimal Control Problem in Hamilton-Jacobi-Bellman Equation Form

Non-linear optimal control problem arises in many different areas, for example, engineering, medical sciences, economics, industries, etc. The solution of Hamilton-Jacobi-Bellman equation is connected with the non -linear optimal control problem. In this paper, we formulate the Hamilton-Jacobi-Bellman equation using nonlinear optimal control problem. We also discuss its solutions using Adomian decomposition method, Laplace transform-Homotopy perturbation method and variational iteration method.

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.

New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

NUMERICAL APPROACHES TO TIME FRACTIONAL BOUSSINESQ–BURGERS EQUATIONS

In this paper, two numerical approaches including the homotopy perturbation transform method (HPTM) and the method based on the fractional complex transform and homotopy perturbation method (named as FCT-HPM) are presented for finding the approximated solutions of the time fractional Boussinesq–Burgers equations with He’s fractional derivative. The FCT-HPM and HPTM solutions are provided without any linearization or complicated computation. Numerical comparisons are shown to illustrate the efficiency of these two methods.

A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

Fractional spatial diffusion of a biological population model via a new integral transform in the settings of power and Mittag-Leffler nonsingular kernel

The current paper examines some novel and interesting aspects of the fractional biological population model involving the efficacious Atangana-Baleanu fractional derivative operator. It assists us in comprehending the dynamical techniques of population changes in biological population models and generates precise prognostication. This technique correlates with the Elzaki transform method and the homotopy perturbation method. The Elzaki transform is a modification of the classical Fourier Laplace transform. The approximate-analytical solutions of the biological population model are examined using the Elzaki transform homotopy perturbation method (ETHPM). The exact solution of the aforesaid scheme is being investigated in terms of the Mittag-Leffler function. The role of fractional-order on spatial diffusion of a biological population model is demonstrated in two and three-dimensional surface plots. The comparative analysis between exact and numerical solutions reveals the innovative features of the composite fractional derivative in the discussed model. Furthermore, the proposed approach is very powerful, reliable, well-organized, and pragmatic for fractional PDEs and it might be extended to other physical processes.

HOMOTOPY PERTURBATION TRANSFORM METHOD FOR SOLVING SYSTEMS OF NONLINEAR PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we apply an efficient method called the Homotopy perturbation transform method (HPTM) to solve systems of nonlinear fractional partial differential equations. The HPTM can easily be applied to many problems and is capable of reducing the size of computational work.

Solution of Non-Linear RLC Circuit Equation Using the Homotopy Perturbation Transform Method

Abstract: In this paper, we apply the Homotopy Perturbation Transform Method (HPTM) to obtain the solution of Non-Linear RLC Circuit Equation. This method is a combination of the Laplace transform method with the homotopy perturbation method. The HPTM can provide analytical solutions to nonlinear equations just by employing the initial conditions and the nonlinear term decomposed by using the He’s polynomials. Keywords: Homotopy perturbation, Laplace transform, He’s polynomials, Non-linear RLC circuit equation.

Qualitative analysis of 2019-nCoV mathematical model via an efficient computational technique

In the present scenario, the whole world struggling with Corona Virus Disease (COVID-19) which is a extremely transmittable disease stimulate by the novel corona virus (2019-nCoV) that believed to be originated in Wuhan, central China. Here, a mathematical analysis of 2019-nCoV mathematical model is considered with Caputo fractional order derivative operator. This model is made up of six subclasses i.e. susceptible people, exposed people, infected people, asymptotically infected people, recovered people and reservoir people. Authors obtained the series solutions of each subclasses of the proposed mathematical model via Sumudu Transform Homotopy Perturbation Method (STHPM). The results that have come are correct to show these authors have provided graphical interpretations.

MATHEMATICAL STUDY OF FRACTIONAL DIABETES MODEL VIA A MODIFIED ANALYTICAL METHOD

In this paper, we obtain the mathematical solution of fractional diabetes model by using Laplace transform homotopy perturbation method (LTHPM). Here the fractional derivatives are well defined in Caputo sense. Numerical simulations are carried out for illustrating the results obtained. In addition, the results obtained by the LTHPM show that the approach is easy to implement and computationally very attractive.

A Reliable Computational Algorithm for Solving Fractional Biological Population Model

In this work, authors obtained the series solution of nonlinear fractional partial differential equations, which is emerging in a spatial diffusion of biological population model using Elzaki transform homotopy perturbation method (ETHPM). The Elzaki transform homotopy perturbation method is a combined form of the Elzaki transform and homotopy perturbation method. Three test illustrations are used to show the proficiency and accuracy of the projected method. It has been observed that the proposed technique can be widely employed to examine other real world problems. The results obtained with the help of the proposed technique are plotted for different fractional orders.

On Black–Scholes option pricing model with stochastic volatility: an information theoretic approach

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic volatility model. Also to investigate the analytical solution of this generalized Black–Scholes equation we have used Laplace transform homotopy perturbation method.

Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations.
 Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method

Very recently, Yang, Abdel-Aty and Cattani (2019) introduced a new and intersting fractional derivative operator with non-singular kernel involving Rabotnov fractional-exponential function. In this paper, we present a general framework of the homotopy perturbation transform method (HPTM) for analytic treatment of time fractional partial differential equations in the sense of Yang-Abdel-Aty-Cattani. As applications, time fractional wave equations involving Yang-Abdel-Aty-Cattani fractional derivatives are solved. The solutions are obtained in the form of series involving Prabhakar functions.

About the Approximate Solutions to Linear and Non-Linear Pseudodifferential Reaction Diffusion Equations

Background. The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function. Objective. To show that the homotopy perturbation transform method (HPTM) is easily applied to linear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations. Methods. The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation. Results. The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series. Conclusions. In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.

Analytical Solution of Ordinary Fractional Differential Equations by Modified Homotopy Perturbation Method and Laplace Transform

In this paper, a modified Laplace transform homotopy perturbation method (MLT-HPM) is used to find the approximate solution of ordinary fractional differential equations(OFDEs). The proposed method is developed to improve the accuracy of the approximate solutions provided by the LT-HPM method. The appropriate initial approximation will be chosen, furthermore, the residual error will be cancelled at several points of the interest interval (RECP). A couple of OFDEs are considered in order to demonstrate the effectiveness of the suggested method. Comparisons are made with the results obtained from LT-HPM, continuous analytical method (CAM), modified homotopy analysis method (MHAM), and modified optimal homotopy analysis method (MOHAM). The resulting solutions require only a first order approximation of MLT-HPM, compared to LT-HPM, which requires more iterations for the same examples studied.

An easy and computable approximation for Troesch’s problem by using the Laplace Transform-Homotopy Perturbation Method

This work introduces the Laplace Transform-Homotopy Perturbation Method (LT-HPM) in order to provide an approximate solution for Troesch’s problem. After comparing figures between exact and approximate solutions, as well as the average absolute relative error (AARE) of the approximate solutions of this research, with others reported in the literature, it can be said that the proposed solutions are accurate and handy. In conclusion, LT-HPM is a potentially useful tool.

Solutions of time-fractional third- and fifth-order Korteweg–de-Vries equations using homotopy perturbation transform method

PurposeThis study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM).Design/methodology/approachHPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials.FindingsThe study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders.Originality/valueAlthough third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.