Numerical solution of nonlinear volterra integral equations using the idea of quasilinearization

The iterations of the quasilinear technique, employed in nonlinear volterra integral equations, are expressed as linear integral equations. By using Collocation Method, the solutions of these linear equations are approximated. Combining this and iterations of the quasilinear technique yields an approximation solution for nonlinear integral equations. The convergence is considered and the examples confirm the accuracy of the solution.     

The scaled boundary FEM for nonlinear problems

The traditional scaled boundary finite-element method (SBFEM) is a rather efficient semi-analytical technique widely applied in engineering, which is however valid mostly for linear differential equations. In this paper, the traditional SBFEM is combined with the homotopy analysis method (HAM), an analytic technique for strongly nonlinear problems: a nonlinear equation is first transformed into a series of linear equations by means of the HAM, and then solved by the traditional SBFEM. In this way, the traditional SBFEM is extended to nonlinear differential equations. A nonlinear heat transfer problem is used as an example to show the validity and computational efficiency of this new SBFEM.     

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

In recent years, many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations. In this paper, the first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including, KdV, Kaup–Boussinesq and Wu–Zhang systems, analytically. By means of this method, some exact solutions for these systems of equations are formally obtained. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

非线性非单调二元算子方程组的解及其应用

利用非线性泛函分析中的锥理论和单调迭代的方法,研究了一类非线性非单调二元算子方程组的解的存在性,并给出了收敛于解的迭代序列,然后作为应用,得到了B anach空间中的一类非线性V olterra型积分方程组的解,改进了最近的许多结果.     

Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach

In this paper, we propose a modified generalized transformation for constructing analytic solutions to nonlinear differential equations. This improved unified ansätze is utilized to acquire exact solutions that are general solutions of simpler equations that are either integrable or possess special solutions. The ansätze is constructed via the choice of an integrable differential operator or a basis set of functions. The technique is implemented to obtain several families of exact solutions for a class of nonlinear evolution equations with nonlinear term of any order. In particular, the Klein–Gordon, the Sine–Gordon and Landau–Ginburg–Higgs equations are chosen as examples to illustrate the method.     

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.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

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.     

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.     

An effective treatment of nonlinear differential equations with linear boundary conditions using the homotopy analysis method

In this paper, we provide an effective technique to treat nonlinear differential equations with linear boundary conditions that are reduced from a nonlinear problem describing the steady-state boundary-layer flow of a micropolar fluid near the forward stagnation point of a two-dimensional plane surface. The analytical approximations with high accuracy are obtained using the homotopy analysis method, which agree well with the numerical results. This indicates the validity and great potential of the proposed method for solving nonlinear differential equations with linear boundary conditions.     

高阶微分积分方程的单调迭代法及其应用

首先利用上下解方法以及微分不等式理论给出了n阶微分积分方程的初值问题解的存在性及其单调迭代法,然后将所得结果应用到n阶微分方程的两点边值问题,得到了n阶非线性两点边值问题解的存在性及其单调迭代法,所得结果推广了已有的结果．     

Stochastic model for the nonlinear point reactor kinetics equations in the presence Newtonian temperature feedback effects

A stochastic model for the nonlinear point reactor kinetics equations with Newtonian temperature feedback and multi-group of precursor delayed neutrons is presented. This model is a couple of the stiff stochastic nonlinear differential equations. The matrix formula of this stochastic nonlinear model is solved by the analytical exponential technique (AET). This proposed technique is based on the integration factor, Euler’s method and the exponential function of the coefficient matrix. This exponential function is determined via the eigenvalues and corresponding eigenvectors of the coefficient matrix. The mean neutron population of the stochastic nonlinear model in the presence Newtonian temperature feedback and six-groups of delayed neutrons is computed for various cases of the external reactivity. The numerical results of the analytical exponential technique are compared with the results of the Euler–Maruyama method and the deterministic results. This comparison confirms that the AET for stochastic nonlinear model is efficient to study the natural behavior of neutron population in the presence temperature feedback effects and multi-group of precursor delayed neutrons.     

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

In this paper, the time fractional partial differential equations are investigated by means of the homotopy analysis method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional partial differential equations.     

非线性算子方程迭代解的存在性定理及其应用

在Ｂａｎａｃｈ空间中,利用锥理论和单调迭代方法研究了一类非线性算子方程的解和最小最大耦合解的存在与迭代逼近定理,并应用到Ｂａｎａｃｈ空间中非线性Ｖｏｌｔｅｒｒａ型积分方程和常微分方程的初值问题．     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

Applications of the extended test approach to (2 + 1)-dimensional Gardner equation

Based on the extended test approach (ETA), we investigated the nonlinear evolution equations, namely, (2 + 1)-dimensional Gardner equation. We aimed to obtain some exact breather-type and periodic-type soliton solutions for this model. These results show that the extended test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations. The properties of some periodic-like and soliton-like solution for this system are shown by some figures.     

Convergence of monotone iterations to initial value problems of functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. We use this technique to initial value problems of functional differential equations showing that corresponding linear iterations converge to the unique solution of our problem and this convergence is superlinear     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.     

The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.