Isogeometric analysis for time-fractional partial differential equations

Numerical Algorithms - In this paper, we study an inverse source problem for the Helmholtz equation from measurements. The purpose of this paper is to reconstruct the salient features of the hidden...     

High-order finite element methods for time-fractional partial differential equations

The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)^2−α+N^−r) is proved for the (r−1)th-order finite element method (r≥2).     

Numerical solutions to time-fractional stochastic partial differential equations

Numerical Algorithms - This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the...     

On the identification of inhomogeneous parameters in dynamic linear partial differential equations

The task of identifying inhomogeneous (position-dependent) coefficients of linear dynamic partial differential equations on the basis of a finite collection of points of the solution has practical importance and is the subject of many published analyses, some of which are described herein. The purpose of the present paper is to present new developments on a simple yet appealing method due to the hydrologist B. Sagar. The technique exploits the viewpoint that the coefficient values of the partial differential equation at any point x are uniquely determined by the solution values in a small neighborhood of x. The identification algorithm which results from these considerations is extremely simple, and yet, in view of technical considerations and experimental evidence set forth here, it seems effective. In particular, we have been able to derive error bounds, which the authors believe is a new feature in the literature of identification of partial differential systems.     

Quantum mechanics and partial differential equations

This paper develops the basic theory of pseudo-differential operators on Rⁿ, through the Calderón-Vaillancourt (0, 0) L²-estimate, as a natural part of the harmonic analysis on the Heisenberg group, the group-theoretic embodiment of Heisenberg's Canonical Commutation Relations. The symbol mapping is given a group-theoretic interpretation consistent with the Kirillov method of orbits. By comparing different well-known realizations of the unique irreducible representation of the Heisenberg group, the Toeplitz operators on the complex n-ball are shown essentially to be pseudo-differential operators. The proof of the Calderón-Vaillancourt estimate is almost purely group-theoretic. Criteria for positivity, and for compactness are also given.     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.     

Numerical solution of fourth-order time-fractional partial differential equations with variable coefficients

In this paper, a numerical method for fourth-order time-fractional partial differential equations with variable coefficients is proposed. Our method consists of Laplace transform, the homotopy perturbation method and Stehfest's numerical inversion algorithm. We show the validity and efficiency of the proposed method (so called LHPM) by applying it to some examples and comparing the results obtained by this method with the ones found by Adomian decomposition method (ADM) and He's variational iteration method (HVIM).     

An effective boundary element method for inhomogeneous partial differential equations

A method for removing the domain or volume integral arising in boundary integral formulations for linear inhomogeneous partial differential equations is presented. The technique removes the integral by considering a particular solution to the homogeneous partial differential equation which approximates the inhomogeneity in terms of radial basis functions. The remainder of the solution will then satisfy a homogeneous partial differential equation and hence lead to an integral equation with only boundary contributions. Some results for the inhomogeneous Poisson equation and for linear elastostatics with known body forces are presented.     

Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics

This review paper gives an extensive overview of the soliton solutions for some famous partial differential equations like KdV, mKdV, Sine–Gordon, and nonlinear Schrödinger equations. Different analytical methods of treatment as well as those of numerical methods are presented in this review paper. Finally relations between the soliton solution and fluid mechanics are shown.     

Differential operators for systems of linear partial differential equations

The aim of this paper is the representation of solutions of systems of formally hyperbolic differential equations of second order. I. N.Vekua gave a representation of the solutions using the Riemann-matrix-function. Here we introduce special differential operators which map holomorphic functions into the set of solutions. An existence theorem for such operators is proved which gives a necessary and sufficient condition on the coefficients of a system. These operators are represented explicitly and several properties of them are investigated. We give different representations of the solutions of such systems and discuss the relation between the integral operator method and the method using differential operators which leads to an explicit representation of the Riemann-matrix-function by means of the differential operators. Two examples of special systems with differential operators are given.     

Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations

Using a new definition of the generalized factorization of linear partial differential operators, we discuss possible generalizations of the Darboux integrability of nonlinear partial differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 144–160, January, 1999.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.The author thanks W. Desch (Graz), I. Gyri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.     

Nonlocal boundary-value problems for systems on linear partial differential equations

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.     

Mean value theorems for solutions of linear partial differential equations

We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with respect to a given vectorM with positive integer components: for each partial derivative occurring in the equation, the inner product of the vector composed of the orders of this derivative in each variable by the vectorM is independent of the derivative. The main results of this paper generalize the well-known Zalcman theorem. Some corollaries are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 260–272, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.