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Chapter 20: Symbolic and Numerical Solutions of ODES with Mathematica
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
The basic idea of finite difference methods is to replace the derivatives in the differential equations by appropriate finite differences. We choose an equidistant grid Xi = a + ih(i=0,…,N+1) $ (i = 0, \ldots , N + 1) $ on [a,b] $ [a, b] $ with step size h=(b-a)/(N+1)(N∈N) $ h = (b - a)/(N + 1)\,\,(N \in {\mathbb{N}}) $ , where X0 = a and XN+1 = b.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
We have discussed in Chapters 5 to 8 that in the finite difference-control volume method the solution domain is divided into a grid of discrete points, called nodes. The governing mathematical equations are then written at each node and its derivatives are expressed by the finite difference formulas, which involve unknown values at discrete grid or nodal points of the domain. This discretization procedure is referred to as point-wise approximation. The system of equations resulting from all nodes including the boundary nodes is solved for the unknown values at the nodal points. The major disadvantages of the finite difference method are: (1) difficulty in accurately representing a geometrically complex domain, (2) difficulty in imposing the boundary conditions along a non-straight boundary, and (3) inability to employ nonrectangular mesh size distribution.
Review of Numerical Methods
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
The finite difference method is a numerical method to approximate derivatives using finite difference equations. For the first derivative approximation, we have the forward difference, the backward difference and the centered difference methods, which can be derived using the Taylor series expansion. The centered difference is the most accurate one in terms of the truncation error. Here, we define four operators to explain the finite different methods in a more general sense, which can also be applied to a set of discrete data. Forward difference: ΔF(xi) = ΔFi = Fi+1 – Fi;Backward difference: ∇F(xi) = ∇Fi = Fi – Fi–1;Stepping: EFi = Fi+1; andDerivative: DFi=dFidx|xi.
Influence of initial stress on shear wave scattering in a functionally graded magneto-visco-elastic orthotropic multi-layered structure
Published in Waves in Random and Complex Media, 2022
Due to the complexity in the analytic calculation, many researchers have developed some novel methods to solve SH-wave propagation in a layered elastic medium. Earlier Thomson [31] and Haskell [32] provided a matrix formalism to the problem of Rayleigh and Love-type waves in multi-layered media. Later, Anderson [33] studied the Love wave propagation in isotropic multi-layer structure using a generalization of Haskell’s technique. Finite difference methods are generally used to solve a problem that is difficult to solve by analytical methods. Chattopadhyay [34] derived closed-form expressions of phase and group velocities in multi-layered structures using Haskell’s matrix method with a finite difference scheme. More recently, Gupta et al. [35] have studied Love wave propagation in a medium composed of multiple orthotropic layers. However, several problems regarding seismic wave propagation in a multi-layered medium have been studied, but in recent times there have not been many works published on functionally graded multi-layered medium.
Analysis for two-dimensional inverse quasilinear parabolic problem by Fourier method
Published in Inverse Problems in Science and Engineering, 2021
The finite difference schemes is considered in this paper. The main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series. The numerical method suggested here is the implicit finite difference method which is second order accurate in the spatial grid sizes and first order in the time grid size. Also the Crank–Nicolson scheme, which is absolutely stable and has a second-order accuracy in the spatial and time grid sizes can be used The Crank–Nicolson scheme require far more computational efforts than the implicit scheme especially in the higher dimensional problems. So the CPU time in the Crank–Nicolson scheme is longer than the implicit finite difference method. The explicit finite difference schemes for the numerical solution of the two-dimensional diffusion equation is the restriction of the size of the time step due to stability requirements.
A dual mesh finite domain method for the numerical solution of differential equations
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
In the finite difference method, the derivatives of the variable of a differential equations are replaced by difference quotients (or the solution variables are expanded in Taylor’s series) that involve the values of the variables at a finite number points of the domain, leading to a formula (called the finite difference “stencil”) whose repeated application yields the final matrix equations among the values of u at the mesh points. The finite difference method, as commonly practiced, suffers from several disadvantages, including inapplicability to non-rectangular domains, inaccurate representation of gradient boundary conditions, inability to use nonuniform and non-rectangular meshes, problem-dependent approximations (i.e., the user cannot select the order of the approximation), and so on. These disadvantages, some of them were overcome in recent years, precluded the development of commercially successful robust, general-purpose, computational tools.