Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

In this paper, we obtain approximate‐analytical solutions of a cancer chemotherapy effect model involving fractional derivatives with exponential kernel and with general Mittag‐Leffler function. Laplace homotopy perturbation method and the modified homotopy analysis transform method were applied. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on homotopy polynomial. The cancer chemotherapy effect equations are solved numerically and analytically using the aforesaid methods. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new fractional‐order derivatives with exponential decay law and with general Mittag‐Leffler law.     

An efficient numerical method for solving coupled Burgers' equation by combining homotopy perturbation and pade techniques

In this study, we combined homotopy perturbation and Pade techniques for solving homogeneous and inhomogeneous two‐dimensional parabolic equation. Also, we apply our combined method for coupled Burgers' equations. The numerical results demonstrate that our combined method gives the approximate solution with faster convergence rate and higher accuracy than using the classic homotopy perturbation method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 982–995, 2011     

Finding the one‐loop soliton solution of the short‐pulse equation by means of the homotopy analysis method

The homotopy analysis method is applied to the short‐pulse equation in order to find an analytic approximation to the known exact solitary upright‐loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Application of variational iteration method for modified Camassa‐Holm and Degasperis‐Procesi equations

In this article, the variational iteration method (VIM) is used to obtain approximate analytical solutions of the modified Camassa‐Holm and Degasperis‐Procesi equations. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The results reveal that the VIM is very effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method

In this article, we apply the homotopy perturbation method (HPM) to obtain approximate analytical solutions of the generalized Burger and Burger‐Fisher (B–F) equations. Several numerical examples are given to illustrate the efficiency of the HPM. Comparison of the result obtained by the present method with exact solution reveals that the accuracy and fast convergence of the new method. It is predicted that the HPM can be found wide application in engineering problems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Application of He's methods for laminar flow in a porous‐saturated pipe

In this paper the problem of fully developed laminar steady forced convection inside a porous‐saturated pipe with uniform wall temperature is presented and the homotopy perturbation method (HPM) and the variational iteration method (VIM) are employed to solve the differential equations governing the problem. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Approximations of the nonlinear Volterra's population model by an efficient numerical method

In this paper, we apply the new homotopy perturbation method to solve the Volterra's model for population growth of a species in a closed system. This technique is extended to give solution for nonlinear integro‐differential equation in which the integral term represents the total metabolism accumulated fromtime zero. The approximate analytical procedure only depends on two components. The newhomotopy perturbationmethodwas applied to nonlinear integro‐differential equations directly and by converting the problem into nonlinear ordinary differential equation. We also compare this method with some other numerical results and show that the present approach is less computational and is applicable for solving nonlinear integro‐differential equations and ordinary differential equations as well. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The numerical solution of the second Painlevé equation

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second‐order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Effect of variable viscosity and viscous dissipation on the Hagen‐Poiseuille flow and entropy generation

In this article, the steady‐state flow of a Hagen‐Poiseuille modelin a circular pipe is considered and entropy generation due tofluid friction and heat transfer is examined. Because of variationin fluid viscosity, the entropy generation in the flow varies. Inhis model, Arrhenius law is applied for temperature equation‐dependent viscosity, and the influence of viscosity parameters on the entropy generation number and distribution of temperature and velocity is investigated. The governing momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by analytical techniques. The solutions of equations via perturbation method and homotopy perturbation method are obtained and then compared with those of numerical solutions. It is found that the fluid viscosity influences considerably the temperature distribution in the fluid close to the pipe wall, and increasing pipe wall temperature enhances the rate of entropy generation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 529–540, 2011     

Application of the homotopy perturbation method to the modified regularized long‐wave equation

In this article, the homotopy perturbation method (HPM) is used to implement the modified regularized long‐wave (MRLW) equation with some initial conditions. The method is very efficient and convenient and can be applied to a large class of problems. In the last, three invariants of the motion are evaluated to determine the conservation properties of the system. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

The travelling wave solutions for non-linear initial-value problems using the homotopy perturbation method

In this article, we have used the homotopy perturbation method (HPM) to find the travelling wave solutions for some non-linear initial-value problems in the mathematical physics. These problems consist of the Burgers–Fisher equation, the Kuramoto–Sivashinsky equation, the coupled Schordinger KdV equations and the long–short wave resonance equations together with initial conditions. The results of these problems reveal that the HPM is very powerful, effective, convenient and quite accurate to the systems of non-linear equations. It is predicted that this method can be found widely applicable in engineering and physics.     

Application of He’s homotopy perturbation method to functional integral equations

In this paper, an application of He’s homotopy perturbation method is applied to solve functional integral equations. Comparisons are made between expansion method based on Lagrange interpolation formulae and the homotopy perturbation method. The results reveal that the He’s homotopy perturbation method is very effective and simple and gives the exact solution.     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.