Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg-de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameter ħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th-order and 14th-order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement

Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.This work was supported in part by the Basque Government, through project IT1207-19

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature

Semi-analytical approach for solving a model for HIV infection of CD4+ T-cells

In this work, a mathematical model for the human immunodeficiency virus (HIV) infection of CD4+ T-cells by using the optimal perturbation iteration method (OPIM) is analyzed. Optimization and classical perturbation techniques are combined to build the new proposed method. The iteration algorithm for systems of nonlinear differential equations for this optimal perturbation iteration technique is constructed for the first time. A test problem has been solved and some plots are given to show the reliability and efficiency of the proposed method. Obtained results exhibit the effectiveness and accuracy of the semi-analytical technique.Publisher's Versio

Approximate analytical solutions of KdV and burgers' equations via HAM and nHAM

This article presents a comparative study of the accuracy between homotopy analysis method (HAM) and a new technique of homotopy analysis method (nHAM) for the Korteweg-de Vries (KdV) and Burgers' equations. The resulted HAM and nHAM solutions at 8th-order and 6th-order approximations are then compared with that of the exact soliton solutions of KdV and Burgers' equations, respectively. These results are shown to be in excellent agreement with the exact soliton solution. However, the result of HAM solution is ratified to be more accurate than the nHAM solution, which conforms to the existing findin

(R1992) RBF-PS Method for Eventual Periodicity of Generalized Kawahara Equation

In engineering and mathematical physics, nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations evidenced by findings link belonging to their long-term actions named as eventual time periodicity discovered over solutions to IBVPs (initial-boundary-value problems). Here we investigate the solution’s eventual periodicity for generalized fifth order Kawahara equation (IBVP) on bounded domain in combination with periodic boundary conditions numerically exploiting mesh-free technique called as Radial basis function pseudo spectral (RBF-PS) method

A series solution for three-dimensional Navier-Stokes equations of flow near an infinite rotating disk

In this paper, homotopy analysis method (HAM) and Padé approximant will be considered for finding analytical solution of three-dimensional viscous flow near an infinite rotating disk which is a well-known classical problem in fluid mechanics. The solution is compared to the numerical (fourth-order Runge-Kutta) solution and the convergence of the obtained series solution is carefully analyzed. The results illustrate that HAM-Padé is an appropriate method in solving the systems of nonlinear equations

Critical flow over an uneven bottom topography using Forced Korteweg-de Vries (fKdV)

This research examined the critical flow over an uneven bump using forced Korteweg-de Vries (fKdV) model. The forced KdV model containing forcing term which represent an uneven bump is solved using Homotopy Analysis Method (HAM). HAM is a semi-analytic technique whereby its solution contains a series of approximated solution in which it converges immediately to the exact solution. A particular HAM solution is chosen with an appropriate convergence parameter by referring to horizontal line segment. The convergent HAM solution depicts that waves only exhibited over the sloping region and no rise of waves found on flat part of bottom topography