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Finite Difference Methods for Hyperbolic PDEs
Published in Victor G. Ganzha, Evgenii V. Vorozhtsov, Numerical Solutions for Partial Differential Equations, 2017
Victor G. Ganzha, Evgenii V. Vorozhtsov
is given. The equation (2.1.2) is called the initial condition for the equation (2.1.1). The mathematical problem (2.1.1)–(2.1.2) is called the initial-value problem, or the Cauchy problem. We now consider the construction of a finite difference scheme for the numerical solution of the initial-value problem (2.1.1)–(2.1.2). In the case of an analytical solution of a PDE, its solution is found in a region of continuous change of the variables x and t. In contrast to the analytic case, the approximate solution of the PDE by the method of finite differences is determined only at some discrete points of the (x,t) plane. If the spatial interval [a1, b1], in which the PDE solution is to be found, is finite, then the number of points in this interval, which is used by the finite difference method, is also finite. Consider a finite set of points in the spatial interval [a1, b1], which satisfies the conditions () a1=x0<x1<x2<…<xM−1<xm=b1
Finite-Difference Schemes for Partial Differential Equations
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
This consideration leads to a very simple explanation of the mechanism of instability. The introduction of a boundary condition merely expands the pool of candidate functions, on which the instability may develop. In the pure von Neumann case, with no boundary conditions, we have only been monitoring the behavior of the harmonics eiαm that are bounded on the entire grid m = 0, ±1, ±2, …. With the boundary conditions present, we may need to include additional functions that are bounded on the semi-infinite grid, but unbounded if extended to the entire grid. These functions do not belong to the von Neumann category. If any of them brings along an unstable eigenvalue |λ| > 1, such as λ = 1 + r/2, then the overall scheme becomes unstable as well. We therefore re-iterate that if the scheme that approximates some Cauchy problem is supplemented by boundary conditions and thus transformed into an initial boundary value problem, then its stability will not be improved. In other words, if the Cauchy problem was stable, then the initial boundary value problem may either remain stable or become unstable. If, however, the Cauchy problem is unstable, then the initial boundary value problem will not become stable.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
Summary for Chapter 4 Since the general solution of a linear differential equation of n-th order contains n arbitrary constants, there are two common ways to specify a particular solution. If the unknown function and its derivatives are specified at a fixed point, we have the initial conditions. In contrast, the boundary conditions are imposed on the unknown function on at least two different points. The differential equation with the initial conditions is called the initial value problem or the Cauchy problem. The differential equation with the boundary conditions is called the boundary value problem.
Traveling waves of nonlocal delayed disease models: critical wave speed and propagation speed
Published in Applicable Analysis, 2023
Hongying Shu, Xuejun Pan, Bruce Wade, Xiang-Sheng Wang
In this section, we develop some numerical algorithms to simulate the traveling wave solutions and estimate the propagation speed of the infectious disease. The theory of traveling wave solutions and propagation speed has been studied extensively in the literature. However, as far as we know, there is not a standard numerical scheme in plotting traveling wave solution and computing the propagation speed. The main challenge lies in the fact that the domain for the boundary value problem (8)–(10) and the Cauchy problem (5)–(10) is infinite. One may suggest truncating the infinite domain on the bounded domain. However, it is a nontrivial task to determine the boundary conditions on the truncated domain. For the traveling wave equations (8)–(10), one may approximate the boundary values of S and I by their limits at . The difficulty comes from the unknown , which, as we shall see later, may be nonzero for some incidence functions. If we impose Neumann condition on the truncated boundaries, then the solutions for the boundary value problem (8)–(10) are not unique because the trivial constant solutions also satisfy the Neumann boundary condition. The truncation method for the Cauchy problem is even more unrealistic because we need to compute the solutions for a sufficiently long time and the truncation error will be significant when the epidemic wave travels to the truncated boundary. To resolve the difficulty, we propose an iteration method to plot the traveling wave solutions and a nonlinear time scale to compute the propagation speed.
A homogenization function technique to solve the 3D inverse Cauchy problem of elliptic type equations in a closed walled shell
Published in Inverse Problems in Science and Engineering, 2021
Chein-Shan Liu, Yaoming Zhang, Fajie Wang
The present numerical solution of the 3D inverse Cauchy problem is to solve an under-specified boundary value problem of an elliptic type partial differential equation (PDE) in a closed walled shell, with the Cauchy data over-specified on an accessible outer boundary. For the purpose of the completion of data, one needs to recover the unknown data on an inaccessible inner boundary of the closed walled shell. However, the inverse Cauchy problem of the elliptic type PDEs is a difficult issue, since the solution does not depend continuously on the given Cauchy data. The errors in the given data may be extremely amplified in the numerical solution, if we cannot get rid of the ill-posedness of the inverse Cauchy problem.
On a strongly continuous semigroup for a Black-Scholes integro-differential operator: European options under jump-diffusion dynamics
Published in Applicable Analysis, 2023
Throughout we assume that , i.e. the space of all bounded and continuous functions endowed with the usual sup norm . Define the operator A whose domain is and such that Thus the initial value problem (7) can be reformulated as an abstract Cauchy problem For the convenience of the reader, let us first recall the definition of a -semigroup and the Hille-Yosida Theorem [14,15].