Class Of Partial Differential Equations Research Articles

Classification of KdV Vessels with Constant Parameters and Two Dimensional Outer Space

In this article we classify vessels producing solutions of some completely integrable partial differential equations (PDEs), presenting a unified approach for them. The classification includes such important examples as Korteweg-de Vries (KdV) and evolutionary non linear Schrodingier (ENLS) equations. In fact, employing basic matrix algebra techniques it is shown that there are exactly two canonical forms of such vessels, so that each canonical form generalize either KdV or ENLS equations. Particularly, Dirac canonical systems, whose evolution was recently inserted into the vessel theory, are shown to be equivalent to the ENLS equation in the sense of vessels. This work is important as a first step to a classification of completely integrable PDEs, which are solvable by the theory of vessels. We note that a recent paper of the author, published in Journal of Mathematical Physics, showed that initial value problem with analytic initial potential for the KdV equation has at least a “narrowing” in time solution. The presented classification inherits this idea and a similar theorem can be easily proved for the presented PDEs. Finally, the results of the work serve as a basis for the investigation of the following problems: (1) hierarchy of the generalized KdV, ENLS equations (by generalizing the vessel equations), (2) new completely integrable PDEs (by changing the dimension of the outer space), (3) addressing the question of integrability of a given arbitrary PDE (the future classification will create a list of solvable by vessels equations, which may eventually include many existing classes of PDEs).

Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions

Abstract We firstly show how one can use the invariant criteria for a scalar linear (1+1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.

On the Solutions of Some Boundary Value Problems for the General Kdv Equation

This paper is concerned with a class of partial differential equations, which are the linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary conditions of a certain special form is studied. We construct some class of solutions of the problem using the inverse spectral method.

О нелокальной краевой задаче для одного класса уравнений с частными производными в цилиндрической области

О нелокальной краевой задаче для одного класса уравнений с частными производными в цилиндрической области

Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems

The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal. <br><br> For more information please click the “Full Text” above.

Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models

AbstractNonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by (where c is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback

It is shown that a strictly-input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain. This follows since the controller imposes exponential dissipation of the total energy. The result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled PDE's and ODE's. The result is specialized to port-Hamiltonian control systems and a simplified DNA-manipulation process is used to illustrate the result.

Generalized semi-analytical solution of advection–diffusion–reaction in finite and semi-infinite cylindrical ducts

Abstract A semi-analytical solution for the transient advection–diffusion–reaction problem within finite and semi-infinite ducts is derived. The solution allows for general radial- and time-dependent inlet/outlet conditions, complex boundary conditions on the duct wall including adsorption and decay, and arbitrary velocity profiles of the transporting fluid. The only numerical step of the solution is the inverse Laplace transform in the time variable. Therefore, the approach also produces fully analytical steady-state solutions. The solution is verified against computational fluid dynamics (CFD) simulations under various boundary conditions and velocity profiles (Newtonian and power-law), and in all cases good agreement is obtained. Although theoretically applicable to all regimes, the solution is computationally difficult at very high Peclet numbers and very early times due to numerical instabilities as a result of finite precision arithmetic of computers. A convergence analysis is conducted to delineate the boundaries of this limit for two important cases. The solution was derived using a new approach for solving two-dimensional partial differential equations (PDEs) with non-constant coefficients which parallels the Frobenius and power series methods for solving ordinary differential equations (ODEs). The approach reduces the original PDE to a single infinite-order ODE with constant coefficients. The approach is suspected to provide solutions to a large class of PDEs of this type. The solution may find applications in a number of engineering and/or biomedical fields, it can be used to verify numerical simulators, and serve as a simple and easy-to-implement alternative where access to numerical simulators is not available.

Reduced-order modelling numerical homogenization.

A general framework to combine numerical homogenization and reduced-order modelling techniques for partial differential equations (PDEs) with multiple scales is described. Numerical homogenization methods are usually efficient to approximate the effective solution of PDEs with multiple scales. However, classical numerical homogenization techniques require the numerical solution of a large number of so-called microproblems to approximate the effective data at selected grid points of the computational domain. Such computations become particularly expensive for high-dimensional, time-dependent or nonlinear problems. In this paper, we explain how numerical homogenization method can benefit from reduced-order modelling techniques that allow one to identify offline and online computational procedures. The effective data are only computed accurately at a carefully selected number of grid points (offline stage) appropriately 'interpolated' in the online stage resulting in an online cost comparable to that of a single-scale solver. The methodology is presented for a class of PDEs with multiple scales, including elliptic, parabolic, wave and nonlinear problems. Numerical examples, including wave propagation in inhomogeneous media and solute transport in unsaturated porous media, illustrate the proposed method.

An Adaptive Radial Point Interpolation Meshless Method for Simulation of Electromagnetic and Optical Fields

This paper is going to handle an adaptive meshless method by which a wide range of microwave problems in terms of frequency are numerically solved whose accuracy and computational time are acceptable with respect to some other numerical schemes. As the origin, in the process of imposing the conventional radial point interpolation method (CRPIM) to laser problems, a special function was found, which results in a well-behaved basis function for CRPIM. This basis function possesses two fundamental advantages in view of meshless methods. At first, and in contradiction with conventional basis functions, the shape parameters are deterministic, which results in a higher accuracy than conventional basis functions. Second, it will construct the shape functions without any need for the middle matrix inversion step. Also, the adaptive basis function inherits the fundamental properties of fields. Hence, the computational time is reduced, approximately by half, comparing with the conventional basis functions. To investigate the proposed adaptive method named quantum radial point interpolation method in different areas of interest, it has been employed to solve three classes of partial differential equations in computational electromagnetics, i.e., Schrodinger's equation in a quantum wave laser, Laplace and electromagnetic wave equations. The results are more accurate and faster than the CRPIM and the finite-element method.

Lattice Differential Equations

The workshop focused on recent advances in the analysis of lattice differential equations such as discrete Klein-Gordon and nonlinear Schrödinger equations as well as the Fermi-Pasta-Ulam lattice. Lattice differential equations play an important role in emergent directions of modern science. These equations are fascinating subjects for mathematicians because they exhibit phenomena, which are not encountered in classical partial differential equations, on one hand, but they may present toy problems for understanding more complicated Hamiltonian differential equations, on the other hand.

Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations

A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted infinitesimal equivalence transformations into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.

Decay Properties for the Numerical Solutions of a Partial Differential Equation with Memory

We study decay properties of the numerical solutions of a class of partial differential equations $$\begin{aligned} \ddot{u}(t)+Au(t)-\int \limits _{0}^{t}\beta (t-s)Au(s)ds=0,\quad t>0,\,u(0)=u_{0...

Adaptive wavelet collocation method on the shallow water model

This paper presents an integrated approach for modeling several ocean test problems on adaptive grids using novel boundary techniques. The adaptive wavelet collocation method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. It is a general method for the solving a large class of partial differential equations, but is applied to the shallow water equations here. In addition to developing wavelet-based computational models, this work also uses an extension of the Brinkman penalization method to represent irregular and non-uniform continental boundaries. This technique is used to enforce no slip boundary conditions through the addition of a term to the field equations. When coupled with the adaptive wavelet collocation method, the flow near the boundary can be well resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow and the boundary is important, thus, those are the test cases presented here.

Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model

A mechanism is presented for the bifurcation from one-dimensional spatially periodic patterns (rolls) into two-dimensional planar states (planforms). The novelty is twofold: the planforms are solutions of a Boussinesq partial differential equation (PDE) on a periodic background and secondly explicit formulas for the coefficients in the Boussinesq equation are derived, based on a form of planar conservation of wave action flux. The Boussinesq equation is integrable with a vast array of solutions, and an example of a new planform bifurcating from rolls, which appears to be generic, is presented. Adding in time leads to a new time-dependent PDE, which models the nonlinear behaviour emerging from a generalization of Eckhaus instability. The class of PDEs to which the theory applies is evolution equations whose steady part is a gradient elliptic PDE. Examples are the 2+1 Ginzburg–Landau equation with real coefficients, and the 2+1 planar Swift–Hohenberg equation.

Multigrid lattice Boltzmann method for accelerated solution of elliptic equations

A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V- or W-cycle) on the performance of the MG–LBM is analyzed. Next, a parallel algorithm for the MG–LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG–LBM.

A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model

The peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by the classical theory of solid mechanics. However, the operator in the peridynamic theory is nonlocal, so the resulting numerical methods generate dense or full coefficient matrices which require O(N2) of memory where N is the number of unknowns in the discretized system. Gaussian types of direct solvers, which were traditionally used to solve these problems, require O(N3) of operations. Furthermore, due to the singularity of the kernel in the peridynamic model, the evaluation and assembly of the coefficient matrix can be very expensive. Numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost! The significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications, especially in multiple space dimensions.We develop a fast and faithful collocation method for a two-dimensional nonlocal diffusion model, which can be viewed as a scalar-valued version of a peridynamic model, without using any lossy compression, but rather, by exploiting the structure of the coefficient matrix. The new method reduces the evaluation and assembly of the coefficient matrix by O(N), reduces the computational work from O(N3) required by the traditional methods to O(Nlog2N) and the memory requirement from O(N2) to O(N). Numerical results are presented to show the utility of the fast method.

Mean Value Theorems for Linear Partial Differential Equations

A symbolic approach is suggested to obtain mean value formulas for solutions of some classes of partial differential equations.

A novel diffusion system for impulse noise removal based on a robust diffusion tensor

The removal of impulse noise is a prerequisite step in image analysis. The classic partial differential equation (PDE) has achieved a great success in suppressing Gaussian noise, but its performance in reducing impulse noise is less satisfactory. The main difficulty arises from finding a nice diffusion function. To tackle this problem, the paper develops a novel diffusion system to suppress impulse noise. The proposed diffusion system consists of two phases. In the first phase, an effective image filter called Clean Pixel Excluder (CPE) is designed to identify clean pixels from the noisy ones. In the second phase, a robust diffusion model is reformulated by developing a novel diffusion tensor to control the smoothing on both direction and strength adaptively. A numerical scheme based on the multi-scale technique is provided. Extensive experiments on both synthetic and real images show that the proposed system achieves a superior performance over several standard methods in terms of noise suppression and detail preservation.

Application of the symmetry classification to the boundary value problem of nonlinear partial differential equations

In this paper, we study the application of the symmetry classification to the boundary value problem of nonli-near partial differential equations. Firstly, by using differential characteristic set algorithm for the complete symmetry classification of partial differential equations, the complete symmetry classification of a given boundary value problem of nonlinear partial differential equations is proposed. Secondly, by using an extended symmetry, the boundary value problem of nonlinear partial differential equations is reduced to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.