Search

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov–Kuznetsov (GZK) equations of form u t + ( F ( u ) ) x x x + ( G ( u ) ) x y y + ( H ( u ) ) x = 0 . As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov–Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations are directly constructed. [ABSTRACT FROM AUTHOR]

In the present paper, Lie symmetries of nonlinear shallow water equations with variable shapes of the bottom that include horizontal, inclined plane and a parabolic bottom are obtained. Exact particular solutions of the governing system are then obtained using the invariance of the system under these symmetries using Lie's method. The evolutionary behaviour of the discontinuity wave, influenced by the amplitude of the discontinuity wave and the geometry of the bottom, is discussed in detail and some contrasting observations are made. [ABSTRACT FROM AUTHOR]

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue. [ABSTRACT FROM AUTHOR]

A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition is studied. It is not a standard heat conduction problem because a non-uniform heat source dependent on the heat flux at the boundary is considered. The purpose of this article is to find explicit solutions and analyze how to control their asymptotic temporal behavior through the source term. Explicit solutions independent of the space or temporal variables, solutions with separated variables and solutions by an integral representation depending on the heat flux at the boundary are given. The controlling effects of the source term are analyzed by comparing the asymptotic temporal behavior of solutions corresponding to the same problem with and without source term. Finally, a relationship between the problem considered here with another non-classical problem for the heat equation is established, and explicit solutions for this second problem are also obtained. In this article, we give explicit solutions and analyze how to control them through the source term for several non-classical heat equation problems. In addition, our results enable us to compute the asymptotic temporal behavior of the heat flux at the boundary for each explicit solution obtained. As a consequence of our study, several solved non-classical problems for the heat equation that can be used for testing new numerical methods are given. [ABSTRACT FROM AUTHOR]

Due to its importance in scientific applications the fractional nonlinear Sharma–Tasso–Olver equation has been studied by several authors from point of view of exact solutions. In this paper, new explicit and exact solutions are obtained by using the fractional complex transform and the improved extended tanh-coth method. The results can be compared with other methods used to solve the same model. [ABSTRACT FROM AUTHOR]

We give an explicit representation of the solutions of the Cauchy problem with analytic data for the following singular partial differential equation(1) We show that the solutions, depending on various parameters, might have singularities on the characteristic surfaces of the operator. [ABSTRACT FROM AUTHOR]

The problem of the immersion of a two-dimensional surface into a three-dimensional Euclidean space can be formulated in terms of the immersion of surfaces in Lie groups and Lie algebras. A general formalism for this problem is developed, as well as an equivalent Mauer–Cartan system of differential forms. The particular case of the Lie group SU (2) is examined, and it is shown to be useful for studying integrable surfaces. Some examples of such surfaces and their equations are presented at the end, in particular, the cases of constant mean curvature and of zero Gaussian curvature. [ABSTRACT FROM AUTHOR]

The propagation of the optical solitons is usually governed by the nonlinear Schrödinger equations. In this article, the two variable $$(\frac{ G^{\prime }}{G},\frac{1}{G})$$ -expansion method is employed to construct the exact traveling wave solutions with parameters of two nonlinear PDEs namely, the (2 $$+$$ 1)-dimensional nonlinear cubic-quintic Ginzburg-Landau equation and the (1 $$+$$ 1)-dimensional resonant nonlinear Schrödinger's equation with dual-power law nonlinearity which describe the propagation of optical pulses in optic fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original $$(\frac{G^{\prime }}{G})$$ -expansion method proposed by M. Wang et al. It is shown that the two variable $$(\frac{G^{\prime }}{G}, \frac{1}{G})$$ -expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics. [ABSTRACT FROM AUTHOR]

In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively skewed stable distributions which produce asymmetric Cauchy densities in the odd-order case. Special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the Painlevé analysis is performed on the physical form of the third-order Burgers' equation, the Painlevé property and integrability (C-integrable) of the equation is verified. Then, the generalized symmetries of the equation are presented and the generalized symmetries of the other equation are given by the symmetry transformation method. The Bäcklund Transformations of the equations are constructed based on the symmetries, respectively. Furthermore, the exact explicit solutions to the equations are investigated in terms of the symmetries, Bäcklund transformations and transformations of the equations. [ABSTRACT FROM AUTHOR]

Monte Carlo acceleration method for pricing variance derivatives under stochastic volatility models with jump diffusion is researched in the paper. Control variate and importance sampling techniques are used to reduce the variance of the simulated price of the derivative. Based on the closed-form solution of a simplified model with piecewise deterministic volatility and jump diffusion, control variate technique is proposed to reduce the simulation errors. Then importance sampling method is also introduced to solve the rare event of the jump part in the model. Through the analysis of the first and second moments of the underlying processes and simplified processes, the method to construct the efficient control variate is proposed. Importance sampling method enhances the effects of the control variate technique. The numerical experiments illustrate the high efficiency of the acceleration method, in accordance with the theoretical analysis. The methods in the paper can also be extended to the pricing of other path-dependent derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the exact solutions to scalar partial differential equations obtained, thanks to the summable Taylor series provided by Adomian's decomposition method. We propose a modification of the method which makes the calculations of Taylor coefficients easier and more direct. The difference is essential, for instance, in the case of non-homogenous equations or initial conditions and is illustrated by some examples. [ABSTRACT FROM AUTHOR]

This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti-Leon-Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this study, homotopy perturbation method (HPM) is used to obtain analytic and approximate solutions of the space- and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of a series with easily computable terms. Some examples are given. The results reveal that HPM is very effective and convenient. [ABSTRACT FROM AUTHOR]

In this article, we first obtain generalized soliton solutions of the general elliptic and auxiliary equations by the Exp-function method. Then by the obtained solutions, we find new and more general solutions of the coupled KdV equation. [ABSTRACT FROM AUTHOR]

The hyperbolic partial differential equations (PDEs) have a wide range of applications in science and engineering. In this article, the exact solutions of some hyperbolic PDEs are presented by means of He's homotopy perturbation method (HPM). The results reveal that the HPM is very effective and convenient in solving nonlinear problems. [ABSTRACT FROM AUTHOR]