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Introduction
Published in Wen L. Li, Weiming Sun, Fourier Methods in Science and Engineering, 2023
Although the method of separation of variables is widely discussed in mathematical physics as a primary means for solving various boundary value problems, its applications are typically limited to the cases when the partial differential equations are linear and homogeneous, and the boundary conditions are homogeneous as well. When a complete system of eigenfunctions is used to expand the solutions, the process of solving the boundary value problems is simplified to seeking a set of expansion coefficients to specially satisfy the differential equation and/or boundary conditions. It is important to point out that such a solution or solution process is valid only when the convergence and effectiveness of the series solution can be confirmed a priori or verified a posteriori.
Vibration of Continuous Systems
Published in William T. Thomson, Theory of Vibration with Applications, 2018
One method of solving partial differential equations is that of separation of variables. In this method, the solution is assumed in the form () y(x,t)=Y(x)G(t)By substitution into Eq. (9.1-2), we obtain () 1Yd2Ydx2=1c21Gd2Gdt2
Function Spaces
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Separation of variables is a technique that may be used to find solutions to some differential equations. The separation we will be mainly concerned with here is the separation of partial differential equations, which essentially amounts to rewriting a differential equation in several variables as an ordinary differential equation in one of the variables and a partial differential equation in the remaining variables. Naturally, if there are only two variables in the problem, the separation will result in two ordinary differential equations. The idea is to reduce the problem to a set of simpler problems that may be solved individually. If we have a partial differential equation in the variables x, y, and z, we can attempt to separate the z coordinate by looking for solutions of the form () f(x,y,z)=g(x,y)Z(z).
Sliding window method for vehicles moving on a long track
Published in Vehicle System Dynamics, 2018
Shuqi Song, Weihua Zhang, Peng Han, Dong Zou
The rail in the sliding window is also modelled with a fourth-order partial differential equation because it is assumed to be a simply supported beam. We can use the separation of variables method to solve the partial differential equations. So, we need to establish the coordinates of the discrete points, which will be used in the mode-superposition method. The origin of the calculation coordinate is set at the left-hand edge of the window and x1 is the coordinate of the first sleeper from the left. By this analogy, the first sleeper coordinate of the nth window is xi and the last one is xj. When the time reaches T+ and the window moves a distance of S1 metres, the original m sleepers are replaced by m new sleepers moving into the window. Thus, the first and last sleeper coordinates at time T+ in the window are xi+m and xj+m, respectively. Because the sleepers and ballast are discrete, the window edge should not be truncated as is the rail. This process can be expressed with the following equations:
Binary gas diffusivity estimates from transient, one-dimensional sublimation–diffusion experiments in a spherical enclosure
Published in Chemical Engineering Communications, 2018
Jeylisse Castañer, Carlos A. Ramírez
Equations (8), (9), (11), and (47) constitute the dimensionless formulation of Case 2. In this situation, since the boundary conditions are not homogeneous, separation of variables cannot be applied directly to Equation (8). However, due to the linearity of the problem, a solution may be postulated which includes a steady-state portion, , satisfying the nonhomogeneous boundary conditions of , and an unsteady portion, , satisfying conveniently defined homogeneous boundary conditions as described next:
Parameter identification for the simulation of the periodontal ligament during the initial phase of orthodontic tooth movement
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Albert Heinrich Kaiser, Ludger Keilig, Reinhard Klein, Christoph Bourauel
A well known method for solving partial differential equations is separation of variables. Analogue, the model function for actuator force is assumed to be the product of a function of actuator displacement and a function of time with time and parameters of F and G: