Using the Natural Generalized Laplace Transform to Solve the Time-Fractional Navier–Stokes Equation

Semi-exact solution of elastic non-uniform thickness and density rotating disks by homotopy perturbation and Adomian's decomposition methods. Part I: Elastic solution

In the present study, we aimed to derive analytical solutions of the homotopy analysis method (HAM) for the time-fractional Navier–Stokes equations in cylindrical coordinates in the form of a rapidly convergent series. In this work, we explore the time-fractional Navier–Stokes equations by replacing the standard time derivative with the Katugampola fractional derivative, expressed in the Caputo form. The homotopy analysis method is then employed to obtain an analytical solution for this time-fractional problem. The convergence of the proposed method to the solution is demonstrated. To validate the method’s accuracy and effectiveness, two examples of time-fractional Navier–Stokes equations modeling fluid flow in a pipe are presented. A comparison with existing results from previous studies is also provided. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics.

Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method

This article presents a new technique for the solution of Time Fractional Navier–Stokes equation. The approach is a combination of Adomian decomposition method (ADM) with Kamal integral transform (KIT). The proposed method is implemented on time-fractional Navier–Stokes equation (TFNSEs) to yield the analytical solution. Illustrative examples in TFNSEs are discussed to validate the applicability of the technique. The graphical visualization of the solutions is also presented. Further, the obtained results are compared with the existing solution methods.

The proof of convergence of the series solution to a class of nonlinear two-dimensional Hammerstein integral equation (NTHIE), including the necessary and su¢ cient conditions that guarantee a unique solution, is introduced. Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM) are used to solve the NTHIE. It was found that, when using the traditional Adomian polynomials (4), ADM and HAM are exactly the same. But, when using the proposed accelerated Adomian polynomials formula (5), ADM converges faster than HAM. The proposed accelerated Adomian polynomials formula is used directly to prove the convergence of the series solution. Convergence approach is reliable enough to estimate the maximum absolute truncated error.

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

This paper considers fifth-order boundary value problems with two-point boundary conditions. For the investigation, first, the Adomian decomposition method (ADM) was applied to the equation and then the homotopy perturbation method (HPM) was used to continue the solution. In addition, this combined method was applied to solve two linear and nonlinear experiments. Then the numerical results obtained by other methods were compared. Furthermore, the convergence of the methods was analyzed. In most of the articles, the HPM is used for equations with initial conditions and the ADM is used for equations with initial and boundary conditions. In this study, a combination of two methods (the ADM and HPM) was used to solve equations with boundary conditions. The results of this combined method are remarkable.

The homotopy analysis method for solving the Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, analytical and numerical solutions for thermoelastic functionally graded (FG) rotating disks with non-uniform thickness under lateral pressure are studied. The study is performed based on Mindlin’s theory. Considering the fact that bending and thermal loadings in analysis of rotating disk are necessary to study the components such as brake and clutch disks. The governing differential equations arising from FG rotating disk are firstly extracted. Then, Liao’s homotopy analysis method (HAM) and Adomian’s decomposition method (ADM) are applied as two analytical approaches. Calculation of stress components and then comparison of the results of HAM and ADM with Runge-Kutta’s and FEM are performed to survey compatibility of their results. The distributions of radial and circumferential stresses of rotating disks are studied and discussed. Finaly, the effects of temperature, grading index, angular velocity and lateral loading on the components of displacement and stresses are presented and discussed, in detail.

<abstract><p>The time-fractional coupled Schrödinger-KdV equation is an interesting mathematical model because of its wide and significant application in mathematics and applied sciences. A fractional coupled Schrödinger-KdV equation in the sense of Caputo derivative is investigated in this article. Namely, we provide a comparative study of the considered model using the Adomian decomposition method and the homotopy perturbation method with Shehu transform. Approximate solutions obtained using the Adomian decomposition and homotopy perturbation methods were numerically evaluated and presented in graphs and tables. Then, these solutions were compared to the exact solutions, demonstrating the simplicity, effectiveness, and good accuracy of the applied method. To demonstrate the accuracy and efficiency of the suggested techniques, numerical problem are provided.</p></abstract>

The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, Adomian decomposition method (ADM) and homotopy analysis method (HAM) are proposed to solving the fuzzy nonlinear Volterra-Fredholm integral equation of the second kind$(FVFIE-2)$. we convert a fuzzy nonlinear Volterra-Fredholm integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. we use ADM , HAM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy Volterra-Fredholm integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed methods are proved. Examples is given and the results reveal that homotopy analysis method is very effective and simple compared with the Adomian decomposition method.

Objectives: This paper obtains the series solution of the cubic complex Ginzburg-Laundau equation, by means of homotopy analysis method(HAM). Methods: In addition to the homotopy analysis method, homotopy perturbation and Adomian decomposition methods are applied to determine approximation solution of the cubic complex Ginzburg-Laundau equation and advantage of using HAM. Also a theorem is proved to guarantee the convergence of the HAM to solve this equation. Findings: Three examples are solved to illustrate the efficiency of the proposed method, this method is compared with other analytical approximate methods such as homotopy perturbation method (HPM)and Adomiam decomposition method(ADM) and it can be seen that these methods have the same results for this equation. Application: Homotopy analysis method as a reliable and valid scheme can be used to work out the cubic complex Ginzburg-Laundau equation which is nonlinear partial differential equation. Keywords: Homotopy analysis method; Ginzburg-Laundau

Revisiting Fisher-KPP model to interpret the spatial spreading of invasive cell population in biology

In this paper, we conduct a comparative study between the homotopy perturbation method (HPM) and Adomian’s decomposition method (ADM) for analytic treatment of nonlinear two-dimensional Volterra-Fredholm fuzzy integral equations (2D-VFFIE) and we show that the HPM with a specific convex homotopy is equivalent to the ADM for these type of equations.