Solutions Of Nonlinear Differential Equations Research Articles

Symbolic computation of analytic approximate solutions for nonlinear differential equations with boundary conditions

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions. In this paper, based on the new definitions of the Adomian polynomials, and the double decomposition method, a new algorithm is proposed to construct analytic approximations for nonlinear differential equations with (coupled) boundary conditions. Furthermore, a MAPLE software package is developed to implement our new algorithm, which is user-friendly and efficient. One only needs to input the system, boundary conditions and several necessary parameters, then our package will deliver the analytic approximate solution within a few seconds. Several different types of nonlinear examples are given to illustrate the scope and demonstrate the validity of our software package, especially for nonlinear multi-order and multi-point boundary value problems. Our package provides a helpful and easy-to-use tool in science and engineering to deal with the most frequently encountered boundary value problems. This MAPLE package is free to download on the website: http://www.cs.ecnu.edu.cn/∼lizb/index.htm.

Existence and uniqueness of asymptotically constant or periodic solutions in delayed population models

By means of fixed point theory we show that the unique solution of nonlinear difference equations of the formconverges to a predetermined constant or a periodic solution. Then, we show the solution is stable and that its limit function serves as a global attractor. The same theory will be extended to two more models.

Generalized homotopy method for solving nonlinear differential equations

In this paper, a new tool for the solution of nonlinear differential equations is presented. The generalized homotopy method (GHM) provides highly accurate approximations of the differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semi-analytic or numerical methods. The obtained results show that GHM is a powerful tool capable to generate highly accurate solutions.

Piece-wise-polynomial method

This work presents the piece-wise polynomial method (PWP) as a novel tool with high potential to find approximate solutions for nonlinear differential equations. Moreover, we present two case studies showing the strength of the method to generate highly accurate approximate solutions. The type of tested nonlinear equations are an asymptotic problem and a boundary value problem.

Global attractivity in a class of non-autonomous, nonlinear, higher order difference equations

Non-autonomous, higher order difference equations of type with real variables and parameters have appeared frequently in the literature. These equations are well defined on Banach algebras, and existing convergence results can be generalized from real numbers to algebras. Through this generalization and by using a recently obtained semiconjugate factorization of the above equation, new sufficient conditions are obtained for the convergence to zero of all solutions of nonlinear difference equations of the above type. Where reduction of order is possible, these conditions extend the ranges of parameters for which the origin is a global attractor even in the case of real variables and parameters.

Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the $ \mathcal{H} $ -classes of these equations.

Nanofluid Flow in a Semi-porous Channel in the Presence of Uniform Magnetic Field

In this paper, the problem of laminar nanofluid flow in a semi-porous channel is investigated analytically using Homotopy Perturbation Method (HPM). This problem is in the presence of transverse magnetic field. Here, it has been attempted to show the capabilities and wide-range applications of the Homotopy Perturbation Method in comparison with the numerical method in solving such problems. The fluid is water containing copper as nanoparticle. The effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell–Garnetts (MG) and Brinkman models, respectively. The obtained solutions, in comparison with the out of the numeric methods admit a remarkable accuracy. A clear conclusion can be drawn from the numerical method’s (NM) results; the mentioned method provides high accurate solutions for nonlinear differential equations. Then, we consider the influence of the three dimensionless numbers: the nanofluid volume friction, Hartmann number for the description of the magnetic forces and the Reynolds number for the dynamic forces. Finally results and discussions are made at the end of this investigation.

On the Effective Region of Convergence of the Decomposition Series Solution

In this paper we investigate the domain of convergence of the Adomian series solution based on the computational results for several examples. We demonstrate how the domain of convergence can be extended by introducing a parameter c in the definition of the zeroth-order and first-order solution components u0 and u1. Furthermore we generalize the concept of the convergence parameter c from a two-term partition of the initial condition to a multiple-term partition with the design of expanding the domain of convergence of the Adomian series solutions for nonlinear differential equations.

On meromorphic solutions of certain type of non-linear differential equations

We consider meromorphic solutions of non-linear differential equation of the form f n + Qd(z;f) = p1(z)e 1(z) + p2(z)e 2(z) ; where Qd(z;f) is a differential polynomial in f of degree d ≤ n − 2 with rational functions as its coefficients, p1;p2 are rational functions and 1; 2 are polynomials. More precisely and mainly we have shown the conditions concerning ' = ' that will ensure the existence and forms of the possible meromorphic solutions of the above equation. These results have extended and improved some known results obtained most recently.

A numerical method for finding soliton solutions in nonlinear differential equations

An iterative method for finding soliton solutions to Korteweg-de Vries and sine-Gordon equations, and for nonlinear Schrodinger equations with cubic nonlinearity, is proposed. In addition, the existence of soliton solutions depending on control parameters is investigated for femtosecond pulse propagation problems in media with cubic nonlinearity. In contrast to other approaches to finding soliton solutions, the proposed method has a high convergence rate, can be applied to multidimensional problems, and does not require special selection of the initial approximation.

Application of GDQ method in nonlinear analysis of a flexible manipulator undergoing large deformation

Analysis of a flexible manipulator as an initial value problem, due to its large deformations, involves nonlinear ordinary differential equations of motion. In the present work, these equations are solved through the general Frechet derivatives and the generalized differential quadrature (GDQ) method directly. The results so obtained are compared with those of the fourth-order Runge–Kutta method. It is seen that both the results match each other well. Further considering the same manipulator as a boundary value problem, its governing equation is a highly nonlinear partial differential equation. Again applying the general Frechet derivatives and the GDQ method, it is seen that the results are in good match with the linear theory. In both cases, the general Frechet derivatives are introduced and successfully used for linearization. The results of the present study indicate that the GDQ method combined with the general Frechet derivatives can be successfully used for the solution of nonlinear differential equations.

A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization

A general algorithm is presented to approximately solve a great variety of linear and nonlinear ordinary differential equations (ODEs) independent of their form, order, and given conditions. The ODEs are formulated as optimization problem. Some basic fundamentals from different areas of mathematics are coupled with each other to effectively cope with the propounded problem. The Fourier series expansion, calculus of variation, and particle swarm optimization (PSO) are employed in the formulation of the problem. Both boundary value problems (BVPs) and initial value problems (IVPs) are treated in the same way. Boundary and initial conditions are both modeled as constraints of the optimization problem. The constraints are imposed through the penalty function strategy. The penalty function in cooperation with weighted-residual functional constitutes fitness function which is central concept in evolutionary algorithms. The robust metaheuristic optimization technique of the PSO is employed to find the solution of the extended variational problem. Finally, illustrative examples demonstrate practicality and efficiency of the presented algorithm as well as its wide operational domain.

On a nonlinear difference equation with variable delay

Abstract In this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delay x n + 1 = λ n x n + α n F ( x n − m n ) , n = 0 , 1 , … $$x_{n + 1} = \lambda _n x_n + \alpha _n F(x_{n - m_n } ),\,\,\,\,n = 0,\,1, \ldots $$ are obtained.

Conservation laws and exact solutions for nonlinear diffusion in anisotropic media

Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.

Some exact solutions to the generalized Korteweg–de Vries equation and the system of shallow water wave equations

In this paper, we establish exact solutions for nonlinear evolution equations in mathematical physics. The exp-transform method is proposed to seek solitary solutions, periodic solutions and compaction-like solutions of nonlinear differential equations. The generalized KdV equation and the system of the shallow water wave equation are chosen to illustrate the effectiveness and convenience of the method.

Variable-Coefficient Exact Solutions for Nonlinear Differential Equations by a New Bernoulli Equation-Based Subequation Method

A new Bernoulli equation-based subequation method is proposed to establish variable-coefficient exact solutions for nonlinear differential equations. For illustrating the validity of this method, we apply it to the asymmetric (2 + 1)-dimensional NNV system and the Kaup-Kupershmidt equation. As a result, some new exact solutions with variable functions coefficients for them are successfully obtained.

Fixed-Term Homotopy

A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature.

Stability Analysis of Two-Segment Stepped Columns with Different End Conditions and Internal Axial Loads

Members with varying geometrical and/or material properties are commonly used in many engineering applications. Stepped columns with internal axial loads constitute a special case of such nonuniform columns. Crane columns in industrial buildings or structural columns supporting intermediate floors are important applications of stepped members in civil engineering. Since neither axial load nor stiffness is constant along the column height, the stability analysis of a stepped column is usually more complicated than that of a uniform column. Determination of exact buckling loads for stepped columns with different end conditions is not always practical. This paper shows that variational iteration method (VIM), a kind of analytical technique recently proposed for solution of nonlinear differential equations, can satisfactorily be used to obtain approximate solutions for buckling loads of stepped columns with internal axial loads. VIM solutions perfectly match with the exact solutions available in the literature for some special cases of two-segment stepped columns. For many other cases, that is, for various values of three design parameters, namely, (i) load ratio, (ii) stiffness ratio, and (iii) length ratio, approximate buckling loads for two-segment stepped columns are determined using VIM and presented in tabular form which can easily be used by design engineers.

Sufficient conditions for existence of solutions of nonlinear differential equations with nonlocal boundary conditions

We provide easily verifiable sufficient conditions for the existence of solutions to non- linear ordinary differential equations subject to nonlocal boundary conditions. These conditions are based on the solution space of the corresponding linear, homogeneous problem and on the size of the nonlinear perturbation. The results presented here are more general nonlinearities than those in (17) and (26).

Time variation of combustion temperature and burning time of a single iron particle

In the present study, combustion of a single iron particle is modeled by virtue of a novel thermophysical procedure. The temperature of iron particle during combustion is studied analytically and numerically. In the proposed model, the effect of thermal radiation from the external surface of burning particle, and alterations of density of iron particle with temperature are considered. Iron particle burns heterogeneously in air. Because of high thermal conductivity and micro size of iron particle, the Biot number is negligibly small, and the lumped system analysis can be utilized for the combustion modeling. The nonlinear energy equation resulted from modeling is solved by using homotopy perturbation method. The assumptions applied in the modeling are such that do not violate the actual combustion phenomenon. Also, the numerical solution of nonlinear differential equation is presented and compared with the analytical solution obtained from homotopy method. It is the first model presented for analyzing the combustion of single iron particle.