C∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.

In this paper a general integral formula for the fundamental solution of a linear partial difference equation with constant coefficients is investigated. For equations in twodimensional space, especially for the discretization of the Helmholtz equation, special integral formulae are given and the convergence of corresponding integrals is proved. The limits of the fundamental solution at infinity are calculated in several cases.

This paper presents a semi-analytical solution for one-dimensional advection-diffusion equation coupled with a linear partial differential equation with constant coefficients. The mathematical model describes a grain-fumigant-air system during fumigation processes, where fumigant gas transports through a storage silo. The coupled system considers both diffusion and advection mechanisms with constant velocity and diffusivity parameters. The solution methodology employs the Laplace transformation technique to convert the partial differential equations into ordinary differential equations in the Laplace domain. The Stehfest numerical algorithm is subsequently applied to invert the Laplace transforms and obtain the time-domain solution. Numerical computations are performed using MATLAB software to simulate the fumigant concentration distributions. Graphical results illustrate the fumigant gas concentration in air versus vertical height within the silo for different time intervals. Additional plots demonstrate the fumigant concentration absorbed by grain particles over time. The analysis examines effects of varying initial gas concentration and flow velocity on the transport process. Results indicate that higher initial concentrations and increased velocities accelerate the fumigation process, requiring less time to fill the silo completely. The proposed solution provides a mathematical framework for optimizing fumigation parameters in agricultural storage applications.

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We treat the master equation in one dimension for the Rayleigh-piston problem of hard rods, i.e., a single heavy tagged particle with finite mass ${m}_{A}$ moving in an ensemble of light bath particles with mass ${m}_{B}$ and a time-independent velocity distribution. The backward form of the master equation is used to obtain a transport equation for a conditional average of a time-dependent physical quantity. It is shown how this integro-differential equation can be solved successively by transforming it into a closed set of first-order linear partial differential equations. The solution of the $l$th linear partial differential equation is completely determined by the solutions of the $l\ensuremath{-}1$ linear partial differential equations, meaning that higher approximations do not change lower ones. It is shown how the motion of the tagged particle can be separated into a deterministic (nonfluctuating) part and into fluctuating contributions. It turns out that the $l=0$ term in the successive approximation scheme satisfies a homogeneous linear partial differential equation and describes the nonfluctuating motion, whereas the higher approximations ($l\ensuremath{\ge}1$), which are solutions of nonhomogeneous linear partial differential equations, describe the fluctuating contributions. The calculations for arbitrary time-dependent conditional averages are performed explicitly up to order 2 in the expansion parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}=\frac{{m}_{B}}{({m}_{A}+{m}_{B})}$. This new method is employed to calculate the conditional averages for the time-dependent mean velocity, the mean-square velocity, the velocity-autocorrelation function, and the self-diffusion coefficient. In addition, the results show that conditional averages calculated via a Gaussian distribution function deviate considerably from the exact results obtained for mass ratios of order 1.

This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.

Moving boundary and boundary value problems occur in many physical and engineering processes involving heat transfer and phase changes, they range from icing problems, for example the Stefan problem, to the melting of metals and plastics using lasers. This book reviews and compares some well known and some recently developed methods to solve these problems. Recent advances in the finite difference solution of linear and non-linear partial differential equations are presented and algorithmic manipulations that enhance the computational efficiency are incorporated into the overall schemes. In this new approach, the the finite difference substitutions are made into the solutions of the partial differential equation rather than classically into the partial differential equation itself. This approach can be exploited for many types of partial differential equation and they may therefore be of interest to researchers studying a wide range of applications in physical problems such as diffusion and heat transfer equations, optoelectronic paraxial wave equations, the quantum mechanical wave equations, time dependent pulse propagation, and similar WKB-derived problemsM

The aim of this research is to study the Z transformation to solve linear partial difference equations. This is due to the fact that linear partial difference equations cannot be solved directly using the Z transformation, and the research concluded that direct partial difference conditions cannot be treated directly by the Z transformation except after converting the partial difference equations to linear difference equations using one of the transformation methods. The research recommended using Binomial transformation to solve partial difference equations.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

On the smoothing property of linear delay partial differential equations

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Recently a new method for solving linear and integrable nonlinear partial differential equations (PDEs) in two dimensions has been introduced. For linear PDEs, this method involves the construction of appropriate integral representations for both the solution and the associated spectral function. Here we present an alternative approach for constructing these integral representations. This approach is based on the introduction of what we call a fundamental differential form, and on a reformulation of Green's formula. The new approach is illustrated for the Laplace and the modified Helmholtz equations in a convex polygon, for a dispersive evolution equation with spatial derivatives of arbitrary order on the half-line, and for the heat equation in a domain involving a moving boundary. In addition we show that the new approach can be used to construct appropriate integral representations for multidimensional linear PDEs.

Stability and asymptotic behavior of solutions for some linear partial functional differential equations in critical cases