Class Of Partial Differential Equations Research Articles

The Picone identity for a class of partial differential equations

The Picone-type identity for the half-linear second order partial differential equation \[ \sum _{i=1}^n\frac{\partial }{\partial x_i} \Phi \bigg (\frac{\partial u}{\partial x_i}\bigg )+c(x)\Phi (u)=0,\quad \Phi (u):=|u|^{p-2}u, p>1, \] is established and some applications of this identity are suggested.

Oscillation of a class of partial difference equations with unbounded delay

This paper is concerned with the partial difference equation ▪ where σ and τ : N → Z are nondecreasing, σ( n) and τ( n) are strictly less than n for n ϵ N, lim n → ∞ σ( n) = lim n → ∞ τ( n) = ∞, { a m,n }, { b m,n }, and { p m,n } are three real double sequences. Some oscillation criteria for this equation are obtained.

A singularity–induced bifurcation theorem for infinite–dimensional implicit dynamical systems

In this note we study a class of partial differential equations whose formal structure resembles the semi–explicit, index–1 form from the theory of differential–algebraic equations. It is known from the theory of such problems that one can encounter simple poles in the eigenvalue loci of parameter–dependent linearizations, leading to the loss of linear stability without the presence of steady–state bifurcations. We demonstrate how this carries over to implicitly defined problems on a Hilbert space.

Symmetries and form-preserving transformations of one-dimensional wave equations with dissipation

General results are given for the forms of infinitesimal point symmetries of the class of partial differential equations in which u tt is a function of x, t, u, u t and the x-derivatives of u up to order n⩾2. Continuous and discrete symmetries are then studied for the more restricted, but wide, class of equations of the form u tt+K(u)u t=(F(u)u x) x+H(u)u x, F(u)≠0 , representing a class of one-dimensional non-linear wave equations with dissipation. A full classification of the groups of continuous point symmetries is given. In addition, new discrete point symmetries are found as well as new transformations which relate equations with different F, H and K.

Invariant set and stable region of a class of partial differential equations with time delays

Invariant set and stable region of a class of partial differential equations with time delays

Input-output stability degrees for undamped constant coefficients linear partial differential equations

It is an established fact that systems which have transfer matrices with poles converging to the imaginary axis cannot have exponentially stable time responses. Recently it was proved that for a class of partial differential equations with such a structure of poles it is possible to have a fast decay in time, uniformly in space, as arbitrary polynomials if the initial conditions are smooth enough and with an appropriate decay at infinity. The non-homogeneous version of this result which we present here can be summarized as follows: 'arbitrary regularity in space of the input function leads to arbitrary polynomial convergence in time towards the steady state'. The relation between the properties of the input function and the rate of convergence of the poles to the imaginary axis is quantitative and we indicate methods for computing this rate. We also provide conditions for exponential stability in this context. Due to some limitations in exponential stabilization by feedback a natural alternative to stability enhancement is this polynomial one. Therefore it is useful to investigate when it can be recovered in more practical situations (bounded space, boundary control, etc). Possible applications include the control of distributed oscillatory phenomena (e.g. in large flexible structures, plates), and more recently the control of some advanced materials.

A numerical study of k(3,2) equation

The Korteweg-de Vries (Kdv) equation has been generalized by Rosenau and Hyman [7] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local conservation laws In this paper, an implicit finite difference method and a finite element method have been developed to solve the K(3,2) equation. Accuracy and stability of the methods have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested methods. The umerical results have shown that this compacton exhibits true soliton behavior.

Pseudo-division algorithm for matrix multivariable polynomial and its application

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective.

Multilayer neural networks for solving a class of partial differential equations

In this paper, training the derivative of a feedforward neural network with the extended backpropagation algorithm is presented. The method is used to solve a class of first-order partial differential equations for input-to-state linearizable or approximate linearizable systems. The solution of the differential equation, together with the Lie derivatives, yields a change of coordinates. A feedback control law is then designed to keep the system in a desired behavior. The examination of the proposed method, through simulations, exhibits the advantages of it. They include easily and quickly finding approximate solutions for complicated first-order partial differential equations. Therefore, the work presented here can benefit the design of the class of nonlinear control systems, where the nontrivial solutions of the partial differential equations are difficult to find.

Viability property for a backward stochastic differential equation and applications to partial differential equations

In the present paper, we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints. This property is the so-called “viability property”. In a separate section, this condition is translated to a class of partial differential equations.

A finite difference method for korteweg-de vries like equation with nonlinear dispersion

The Korteweg-de Vries (KdV) equation has been generalized by Rosenau and Hyman [3] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons. Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local consevation laws. In this paper, an implicit numerical method has been developed to solve the K(2,3) equation. Accuracy and stability of the method have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested method. The numerical results have shown that this compacton exhibits true soliton behavior.

Rm _ type solutions for a class of partial differential equations

Some flmdamental Solutions of r”*- type for a dass of ıterated eUiptic equations are given, inciuding Laplace equation and its iterates.

Some maximum principles for solutions of a class of partial differential equations in Ω ⊂ ℝn

We find maximum principles for solutions of semilinear elliptic partial differential equations of the forms: (1)Δ2u+αf(u)=0,α∈ℝ+and (2)ΔΔu+α(Δu)k+gu=0,α≤0in some regionΩ⊂ℝn.

The Classification of Bifurcations with Hidden Symmetries

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an N-dimensional rectangle under Neumann boundary conditions.

A Picard‐Maclaurin theorem for initial value PDEs

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on ℜn with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a solution of a PDE is a subset of ℜn, we identify one component of the domain to achieve the analogy with ODEs. The generator for the PDE must be a polynomial and autonomous with respect to this component, and no partial derivative with respect to this component can appear in the domain of the generator. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. As in the ODE case these conditions can be met, for a broad class of PDEs, through polynomial projections.

Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.

AN INTEGRATION SCHEME FOR REACTION–DIFFUSION MODELS

A detailed description and validation of a recently developed integration scheme is here reported for one- and two-dimensional reaction–diffusion models. As paradigmatic examples of this class of partial differential equations the complex Ginzburg–Landau and the Fitzhugh–Nagumo equations have been analyzed. The novel algorithm has precision and stability comparable to those of pseudo-spectral codes, but is more convenient to be employed for systems with large linear extention L. As for finite-difference methods, the implementation of the present scheme requires only information about the local enviroment and this allows us to treat systems with very complicated boundary conditions.

A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow

The advection-diffusion equation is studied via a global Lagrangian coordinate transformation. The metric tensor of the Lagrangian coordinates couples the dynamical system theory rigorously into the solution of this class of partial differential equations. If the flow has chaotic streamlines, the diffusion will dominate the solution at a critical time, which scales logarithmically with the diffusivity. The subsequent rapid diffusive relaxation is completed on the order of a few Lyapunov times, and it becomes more anisotropic the smaller the diffusivity. The local Lyapunov time of the flow is the inverse of the finite time Lyapunov exponent. A finite time Lyapunov exponent can be expressed in terms of two convergence functions which are responsible for the spatio-temporal complexity of both the advective and diffusive transports. This complexity gives a new class of diffusion barrier in the chaotic region and a fractal-like behavior in both space and time. In an integrable flow with shear, there also exist fast and slow diffusion. But unlike that in a chaotic flow, a large gradient of the scalar field across the KAM surfaces can be maintained since the fast diffusion in an integrable flow is strictly confined within the KAM surfaces.

Computability structure of the Sobolev spaces and its applications

In this paper we present a definition of s-computability for functions in Sobolev spaces H s ( R d ), s ϵ R. Two applications of s-computability are then discussed. One is to address several open problems regarding classical computable functions; the other is to conduct computable analysis of several important classes of partial differential equations.

Exact analytic solutions to turbulent particle flow equations

A technique for solving a certain class of partial differential equations is used to obtain exact solutions for particle dispersion in an unbounded Gaussian random flow field in which the mean flow is a simple shear and the turbulence is homogeneous and stationary. Results are compared with those of a simple random walk simulation and those obtained from tracking particles in a simple developing shear generated by LES.