Nonlinear Equations Of Mathematical Physics Research Articles

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp (−φ(η))-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (−φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.

Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity

We consider an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Under certain conditions on the operators in the equation, we prove its local (in time) solubility and give sufficient conditions for finite-time blow-up of solutions of the corresponding abstract Cauchy problem. The proof uses a modification of a method of Levine. We give examples of Cauchy problems and initial-boundary value problems for concrete non-linear equations of mathematical physics.

Symbolic Computation and the Extended Hyperbolic Function Method for Constructing Exact Traveling Solutions of Nonlinear PDEs

On the basis of the computer symbolic system Maple and the extended hyperbolic function method, we develop a more mathematically rigorous and systematic procedure for constructing exact solitary wave solutions and exact periodic traveling wave solutions in triangle form of various nonlinear partial differential equations that are with physical backgrounds. Compared with the existing methods, the proposed method gives new and more general solutions. More importantly, the method provides a straightforward and effective algorithm to obtain abundant explicit and exact particular solutions for large nonlinear mathematical physics equations. We apply the presented method to two variant Boussinesq equations and give a series of exact explicit traveling wave solutions that have some more general forms. So consequently, the efficiency and the generality of the proposed method are demonstrated.

Von mises- and crocco-type hydrodynamical transformations: Order reduction of nonlinear equations, construction of Bäcklund transformations and of new integrable equations

Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Backlund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.

Modified Newton, rank-1 Broyden update and rank-2 BFGS update methods in helicopter trim: A comparative study

Nonlinear equations in mathematical physics and engineering are solved by linearizing the equations and forming various iterative procedures, then executing the numerical simulation. For strongly nonlinear problems, the solution obtained in the iterative process can diverge due to numerical instability. As a result, the application of numerical simulation for strongly nonlinear problems is limited. Helicopter aeroelasticity involves the solution of systems of nonlinear equations in a computationally expensive environment. Reliable solution methods which do not need Jacobian calculation at each iteration are needed for this problem. In this paper, a comparative study is done by incorporating different methods for solving the nonlinear equations in helicopter trim. Three different methods based on calculating the Jacobian at the initial guess are investigated.

On order reduction of non-linear equations of mechanics and mathematical physics, new integrable equations and exact solutions

Some classes of non-linear equations of mechanics and mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where a first-order partial derivative is taken as a new independent variable and a second-order partial derivative is taken as the new dependent variable. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes, Euler, and boundary layer) and deriving exact solutions to these equations. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases include Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics). A number of new integrable non-linear equations, inclusive of the generalized Calogero equation, are considered.

On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.

Study and application of optimization algorithm about nonlinear partial differential equations with boundary value problem

For elliptic nonlinear partial differential equations with boundary value problem, based on difference method and dynamic design variable optimization method, by taking unknown function value on discrete net point as design variables, difference equation of all the discrete net points is constructed as an objective function. A kind of optimization algorithm about solving unknown function value on discrete net point is proposed. Universal computing program is designed. Practical example is analyzed. By comparing the computing result with the analytical solution, effectiveness and feasibility are verified. Thus complicated nonlinear mathematical physics equations can be solved by the numerical calculation method.

Using [formula omitted]-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii–Piskunov equation

In this paper, we use the G′G-expansion method to seek the traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation. The solutions obtained in this paper are more general than the solutions given in Refs. [23–25], and the computation procedure is much simpler. It is shown that the G′G-expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics.

A simple direct method to find equivalence transformations of a generalized nonlinear Schrödinger equation and a generalized KdV equation

A simple direct method is presented to find equivalence transformations of nonlinear mathematical physics equations. By using the direct method, we obtain the continuous equivalence transformations of a class of nonlinear Schröequations with variable coefficients and a family of nonlinear KdV equations with variable coefficients. For the nonlinear Schrödinger equations with variable coefficients, the equivalence transformations obtained by the direct method coincide, in nature, with those obtained via the infinitesimal Lie criterion, but our computation is much simpler.

Exact solutions for some non-linear differential equations using complex hyperbolic function methods

Based on computerized symbolic computation, the coth–csch complex hyperbolic-function method is proposed for the general non-linear equations of mathematical physics in a unified way. In this article, we assume that exact solutions for a given general non-linear equations be the superposition of different powers of the coth-function, csch-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general non-linear equations directly without transformation. Some illustrative equations, such as the coupled non-linear Schrödinger equation, the generalized non-linear Schrödinger-like equation, coupled non-linear Schrödinger KdV system, Davey Stewartson equation and the (2 + 1)-dimensional generalization of coupled non-linear Schrödinger KdV system equation are investigated by this method and new exact solutions are found.

Solution of ODE u″ + p(u)(u′)2 + q(u) = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc, are reduced to an integrable ODE expressed by u″ + p(u)(u′)2 + q(u) = 0 whose general solution can be given. Furthermore, combining complete discrimination system for polynomial, the classifications of all single travelling wave solutions to these equations are obtained. The equation u″ + p(u)(u′)2 + q(u) = 0 includes the equation (u′)2 = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.

A new method applied to obtain complex Jacobi elliptic function solutions of general nonlinear equations

Based on computerized symbolic computation, a new complex Jacobi elliptic function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equation be the superposition of different powers of the Jacobi elliptic function. By finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex Jacobi elliptic function. The characteristic feature of this method is that, without transformation, we can derive exact solutions to the general nonlinear equations directly. Some illustrative equations, such as the (1 + 1)-dimensional Klein–Gordon–Zakharov equation, (2 + 1)-dimensional long-wave–short-wave resonance interaction equation, are investigated by this means to find the new exact solutions.

Non-travelling wave solutions of two (2 + 1)-dimensional nonlinear mathematical physics equations

The overall objective of this paper is to find and analyze the new nonlinear travelling wave solutions of two nonlinear mathematical physics equations: the (2 + 1)-dimensional Perturbed-KdV equation and the (2 + 1)-dimensional breaking soliton equation. In this paper, based on symbolic computation and the further improved F-expansion method, many new non-travelling wave solutions of the two equations are obtained and their properties are studied by analyzing their figures.

Study of the plane problem for a physically nonlinear elastic solid by methods of the theory of functions of one complex Variable

Successful application of methods of complex analysis in linear elasticity problems, initiated by Kolosov, Muskhelishvili, Vekua, and their students, serves as a basis for similar studies in the field of analytical-numerical approximations to solutions of boundary value problems and various nonlinear equations of mathematical physics. In the present paper, we suggest a method for solving plane boundary-value problems for a special class of physically nonlinear elastic solids in the case of small strains. This general method, which can be used for a wide class of domains, is illustrated by the example of a square domain with boundary conditions given in stresses. These methods can also easily be used for boundary conditions of other types.

Complex hyperbolic-function method and its applications to nonlinear equations

Based on computerized symbolic computation, a complex hyperbolic-function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equations be the superposition of different powers of the sech-function, tanh-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general nonlinear equations directly without transformation. Some illustrative equations, such as the ( 1 + 1 )-dimensional coupled Schrödinger–KdV equation, ( 2 + 1 )-dimensional Davey–Stewartson equation and Hirota–Maccari equation, are investigated by this means and new exact solutions are found.

Generalized method to construct the solitonic solutions to [formula omitted]-dimensional nonlinear equation

A generalized method with computerized symbolic computation, which is called the generalized extended tanh-function method, is presented for the nonlinear equations of mathematical physics (NEMPs). Compared with most of the existing tanh-function method, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEMPs. Make use of the method, we study the ( 3 + 1 ) -dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the nontraveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions.

Inverse problems for linear and non-linear equations of mathematical physics

Inverse problems for linear and non-linear equations of mathematical physics

Inverse problems for linear and non-linear equations of mathematical physics

In the paper we consider inverse problems for evolutionary linear and non-linear equations. We introduce constructive methods for their investigation, in particular, in some cases of the formula.

New expansion algorithm of three Riccati equations and its applications in nonlinear mathematical physics equations

Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schrodinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions and singular solutions. Some new solutions are illustrated in figures.