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This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.
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19Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay
Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay
8New Exact Solutions of (1 + 1)-Dimensional Coupled Integrable Dispersionless System
This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more general variable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) in our solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.
Read more18Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients
Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients
Read more38The exact solution of a linear integral equation with weakly singular kernel
The exact solution of a linear integral equation with weakly singular kernel
8Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations
We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.
Read more4Exact Solutions of a Nonclassical Nonlinear Equation of the Fourth Order
For a nonclassical nonlinear partial differential equation of the fourth order, ten families of exact solutions expressed in terms of elementary and special functions are constructed. These results are compared with previously obtained exact solutions of the equation.
Read more39Traveling wave and exact solutions for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity
The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.
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Author index
Exact Solutions of the Nonlinear Equations for Self-Gravitating Matter
An important problem in the theories of galaxy formation is the evolution of density inhomogeneities in the regime where perturbation theory fails. Here, several exact solutions of the equations of Newtonian hydrodynamics for self-gravitating matter are discussed, that provide models for the “pancake”-collapse, for flat, one-dimensional structures with non-vanishing gas pressure and for a “pancake”-like collapse with pressure in two dimensions: The general exact solution for vanishing pressure and two vanishing eigenvalues of the expansion tensor is derived, modeling a highly anisotropic collapse on large scales. It is also shown that all nonlinear plane-wave solutions in an isothermal gas can be generated by a sine-Gordon equation, leading to soliton solutions for the gravity system. A two-dimensional solution with gas pressure is obtained by conformai transformations from the hydrostatic equilibrium showing a “pancake”-like collapse on certain hypersurfaces which are determined by the nodal lines of a two-dimensional Laplace equation.
Read more29Rational Solutions of Riccati-like Partial Differential Equations
Rational Solutions of Riccati-like Partial Differential Equations
15Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations
Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations
1A Very Simple Procedure for Constructing the Solution of Nonlinear Elliptic PDEs (with Nonlinear Boundary Conditions) Arising from Heat Transfer Problems in Solids
This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, the partial differential equations which govern the steady-state heat transfer process in bodies with temperature-dependent thermal conductivity and temperature-dependent internal heat source. In addition, it will be considered a large class of nonlinear boundary conditions, especially those arising from the description of the heat exchange process from/to bodies at high temperature levels. Proofs of the solution’s existence and uniqueness are presented.
Read more6Exact solutions of a novel integrable partial differential equation
Exact solutions of a novel integrable partial differential equation
2Faber-Schauder Wavelet Sparse Grid Approach for Option Pricing with Transactions Cost
Transforming the nonlinear Black-Scholes equation into the diffusion PDE by introducing the log transform ofSand(T−t)→τcan provide the most stable platform within which option prices can be evaluated. The space jump that appeared in the transformation model is suitable to be solved by the sparse grid approach. An adaptive sparse approximation solution of the nonlinear second-order PDEs was constructed using Faber-Schauder wavelet function and the corresponding multiscale analysis theory. First, we construct the multiscale wavelet interpolation operator based on the definition of interpolation wavelet theory. The operator can be used to discretize the weak solution function of the nonlinear second-order PDEs. Second, using the couple technique of the variational iteration method (VIM) and the precision integration method, the sparse approximation solution of the nonlinear partial differential equations can be obtained. The method is tested on three classical nonlinear option pricing models such as Leland model, Barles-Soner model, and risk adjusted pricing methodology. The solutions are compared with the finite difference method. The present results indicate that the method is competitive.
Read more2Exact solution of the Bethe-Salpeter equation with retarding propagators in the Wick-Cutkosky model
We study the discrete spectrum of the Bethe-Salpeter equation in the Wick-Cutkosky model and show that for retarding propagators, the variables can be separated, and in the S-wave case, the equation can be reduced to a one-dimensional equation. By using the method of contour integration, we obtain the exact solution of this equation and study the spectrum of relativistic bound states. We discuss the ambiguity of the analytic expression of the wave function that is caused by the presence of a nonphysical variable in the four-dimensional description of relativistic systems.
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