On the smoothing property of linear delay partial differential equations

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 work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.

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.

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.

Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM)

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.

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.

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.

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.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

A Lyapunov direct method is proposed for a class of underactuated, mechanical systems. The direct method is derived in general for systems having n degrees of freedom of which only m < n are actuated. The applications consist of a class of systems where the elements of the mass/inertia matrix and the gravitational forces/torques are either constants or functions of a single generalized position variable and where n is two and m is one. The time derivative of the candidate Lyapunov function produces a relation that is solved via a matching method. Some of the matching equations consist of linear differential and partial differential equations. It is shown for this class of systems, that the solutions of these linear differential and partial differential equations necessary for assuring asymptotic stability can be evaluated numerically as part of the feedback process. Examples are presented involving an inverted pendulum cart and an inertia wheel pendulum.

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley &amp; Sons, Ltd.

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\)(E) of the space ℰ(E) of the Silva C ∞ functions on E is defined; the dual of \(\mathbb{E}\)(E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\)(E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite-dimensional system. Applications of the linear dynamic output feedback controller to the linearized stochastic Kuramoto-Sivashinsky equation are presented.

There are no strict rules for the numerical treatment of partial differential equations. We concentrate here on linear partial differential equations. As an example of elliptic partial differential equations the two-dimensional Poisson equation with Dirichlet boundary conditions is investigated. The application of the finite difference approximation transforms this equation into a set of algebraic equations which can be solved iteratively. The time dependent heat equation is an example of parabolic partial differential equations. The one-dimensional Dirichlet boundary value problem is transformed using the implicit and explicit Euler method into an inhomogeneous system of linear algebraic equations. Stability problems of the explicit Euler method are discussed and demonstrated numerically. The wave equation is a model for hyperbolic partial differential equations. It can be transformed into a system of algebraic equations using, for instance, the explicit Euler method. Again, stability problems connected with this method are discussed and demonstrated numerically. Finally, the quantum mechanical tunneling effect is studied numerically by solving the time dependent Schrodinger equation by means of a Crank-Nicholson scheme which transforms the Schrodinger equation into a system of linear algebraic equations with tridiagonal matrix.