Third-order Partial Differential Equation Research Articles

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

Explicit solutions of some linear ordinary and partial fractional differintegral equations

In recent years, many authors demonstrated the usefulness of some fractional calculus operators in the derivation of (explicit) particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how readily several recent contributions on this subject, involving second- as well as third-order linear ordinary and partial differential equations, can be obtained (in a unified manner) by appropriately applying some general theorems on (explicit) particular solutions of a certain family of linear ordinary and partial fractional differintegral equations.

An analytic study on the third-order dispersive partial differential equations

In this paper we propose a new approach for solving the third-order dispersive partial differential equations in one- and higher-dimensional spaces. Our approach depends mainly on Adomian decomposition method. The modified decomposition method and the noise terms phenomenon will be used for non-homogeneous problems. We demonstrate our work by testing numerical examples designed to capture the features of the proposed algorithms.

MHD flow of a third-grade fluid due to eccentric rotations of a porous disk and a fluid at infinity

The problem of magnetohydrodynamics (MHD) flow of a conducting, incompressible third-grade fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is considered. An exact analysis is carried out to model the governing non-linear partial differential equation. A numerical solution of the third-order non-linear partial differential equation has been obtained. Several graphs and tables have been drawn to show the influence of porosity ε, magnetic parameter N, material parameters α and β on the velocity distribution.

Analysis and classification of nonlinear dispersive evolution equations in the potential representation

A potential representation for the subset of travelling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves reduction of a third-order partial differential equation to a first-order ordinary differential equation. The potential representation allows us to deduce certain properties of the solutions without the actual need to solve the underlying evolution equation. In particular, the paper deals with the so-called K(n, m) equations. Starting from their respective potential representations it is shown that these equations can be classified according to a simple point transformation. As a result, e.g., all equations with linear dispersion join the same equivalence class with the Korteweg–deVries equation being its representative, and all soliton solutions of higher order nonlinear equations are thus equivalent to the KdV soliton. Certain equations with both linear and quadratic dispersions can also be treated within this equivalence class.

Magnetic fluctuations with a zero mean field in a random fluid flow with a finite correlation time and a small magnetic diffusion.

Magnetic fluctuations with a zero mean field in a random flow with a finite correlation time and a small yet finite magnetic diffusion are studied. Equation for the second-order correlation function of a magnetic field is derived. This equation comprises spatial derivatives of high orders due to a nonlocal nature of magnetic field transport in a random velocity field with a finite correlation time. For a random Gaussian velocity field with a small correlation time the equation for the second-order correlation function of the magnetic field is a third-order partial differential equation. For this velocity field and a small magnetic diffusion with large magnetic Prandtl numbers the growth rate of the second moment of magnetic field is estimated. The finite correlation time of a turbulent velocity field causes an increase of the growth rate of magnetic fluctuations. It is demonstrated that the results obtained for the cases of a small yet finite magnetic diffusion and a zero magnetic diffusion are different.

A General Darboux-Type Boundary Value Problem in Curvilinear Domains with Corners for a Third-Order Equation with Dominated Lower-Order Terms

We study solvability of the Darboux-type boundary value problem for a third-order linear partial differential equation with dominated lower-order terms. We indicate function spaces in which the problem is uniquely solvable and Hausdorff normally solvable. In the second case, the corresponding homogeneous problem is shown to have infinitely many linearly independent solutions.

Similarity solutions to non-linear partial differential equation of physical phenomena represented by the Zakharov–Kuznetsov equation

The non-linear partial differential equation entitled the Zakharov–Kuznetsov equation which governs the problem of ion-acoustic waves in a magnetized plasma, has been analyzed via symmetry method as developed by S. Steinberg [Symmetry methods in differential equations, Technical report no. 367, University of New Mexico, 1979]. It is shown that the symmetries form a finite dimensional vector Lie algebra. The carryover of the symmetry method has led to certain similarity reductions of this equation to a solvable form of third-order partial differential equations, depending on certain choices of the parameters occurring in the symmetries. The infinitesimals, similarity variables, dependent variables, and reductions to quadrature or exact solutions have been tabulated. The outcomes of this study are quite in the sense of the deduction of some new exact solutions that does not seem to have been exploited.

Asymptotic representations for a class of multiple hypergeometric functions with a dominant variable

Motivated essentially by several recent studies of the Goursat-Darboux problems for third-order partial differential equations, asymptotic representations are obtained here for some multiple hypergeometric functions (a type of generalized Lauricella functions) when one of the variables tends to infinity and the others tend to zero. Several examples illustrating the applicability of the main theorems are considered and an open problem relevant to the present investigation is also indicated.

Self-Sustained Oscillations of Nonlinearly Viscoelastic Layers

We study the motion of an incompressible nonlinearly viscoelastic layer subject to a slip-stick frictional force applied to one of its faces by a moving belt. This system is governed by a third-order quasi-linear parabolic-hyperbolic partial differential equation subject to complicated boundary conditions. We use a refined version of the Hopf bifurcation theorem to show that this problem admits periodic solutions for the belt speed near critical values, with the number of such solutions increasing as the viscosity of the layer decreases. To verify the hypotheses of this theorem, we must devote considerable effort to the tricky analysis of how the spectrum of the linearized problem depends on the belt speed and material parameters. Our analysis is complemented with both computations and a proof of the topological fact that the computed disposition of the eigenvalues does not omit any others. We complete our study with a perturbation analysis, justified by the Hopf bifurcation theorem.

Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

We prove existence and uniqueness results for nonlinear third-order partial differential equations of the form

Perturbation solution for small amplitude solitary waves in two-phase fluid flow of compacting media

A perturbation solution for small-amplitude solitary waves is derived for the third-order nonlinear partial differential equation due to Scott and Stevenson which describes the one-dimensional migration of melt through the Earth's mantle. The straightforward perturbation expansion breaks down and a coordinate stretching transformation is performed to render the perturbation expansion uniformly valid. The lowest-order perturbation solution has the same form as the single-soliton solution of the Korteweg-de Vries equation. The perturbation solution is derived to second order in implicit form. It is found to be extremely accurate when compared with known exact solutions for specific values of the exponents n and m. The zero- and first-order perturbation solutions are found to be accurate when n = m. The properties of small-amplitude solitary waves are investigated using the perturbation solution.

The Spectrum of Coupled Random Matrices

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices. Comment: 56 pages, published version, abstract added in migration

In Situ Determination of Soil Stiffness and Damping

Determination of in situ dynamic soil properties is fundamental to the prediction of the seismic behavior of foundations and soil embankment structures. Both elastic (stiffness) and inelastic (damping) values are required for computational analysis. To be of value to engineers, the geophysical inversion should employ the same soil model as used in the dynamic analysis software. Current engineering practice employs a Kelvin-Voigt model (spring in parallel with dashpot). The relevant wave equation is a third-order partial differential equation. This paper demonstrates how to collect in situ field data and solve for stiffness (scaled shear) and damping values by a method consistent with this constitutive model. Measurements of seismic wave amplitude decay and velocity dispersion are simultaneously inverted for the required stiffness and damping values. These in situ stiffness and damping values are directly comparable to those obtained by resonant column measurements in the laboratory. Furthermore, the results may be directly input into currently available engineering software to provide values of stiffness and viscous damping. This paper includes both synthetic (finite difference) and field data examples that illustrate the method.

A first-second order splitting method for a third-order partial differential equation

A splitting of a third order partial differential equation into a first-order and a second-order one is proposed as the basis for a mixed finite element method to approximate its solution. A time-continuous numerical method is described and error estimates for its solution are demonstrated. Finally, a full discretization is described based on backward Euler finite differences in time, and error estimates for the resulting approximation are established. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 89–96, 1998

Symmetries of a class of nonlinear third-order partial differential equations

In this paper, we study symmetry reductions of a class of nonlinear third-order partial differential equations u t − ϵu xxt + 2 κu x = uu xxx + αuu x + βu x u xx , (1) where ϵ, κ, α, and β are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case, the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters ϵ = 1, α = −1, β = 3, and κ = 1 2 , admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters ϵ = 0, α = 1, β = 3, and κ = 0, admits a “compacton” solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters ϵ = 1, α = − 3, and β = 2, has a “peakon” solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.

Evaluation of Chebyshev pseudospectral methods for third order differential equations

When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and extreme eigenvalue already reaching O(N 5 for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N 4. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach.

A Few Thoughts About Quarks

We present a novel approach to the description of quark fields, based on the use of Z_{3}-graded algebras. After introducing Z_{3}-graded generalizations of Grassmann and Clifford algebras, we discuss the cubic version of Dirac's equation for the quarks, which after diagonalization leads to the third-order partial differential equation for the wave functions. We show how certain cubic and quadratic products of these functions can be interpreted as solutions of first-and second-order differential equations.

Mathematical Simulation of Heat Transfer in Relaxing Media

We find a numerically-analytical solution of a boundary problem for the third-order partial differential equation, which describes the mass and heat transfer in active media.

Error Estimates for Finite Difference Methods for a Wide-Angle “Parabolic” Equation

We consider a model initial and boundary value problem for a third-order partial differential equation (PDE), a wide-angle parabolic equation frequently used in underwater acoustics, with depth- and range-dependent coefficients in the presence of horizontal interfaces and dissipation. After commenting on the existence--uniqueness theory of solution of the equation, we discretize the problem by a second-order finite difference method of Crank--Nicolson type for which we prove stability and optimal-order error estimates in suitable discrete L2-, H1-, and maximum norms. We also prove, under certain conditions, that the forward Euler scheme is also stable and convergent for the problem at hand.