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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
A singular solution of special interest is one that consists entirely of branch points—at every point it is tangent to another integral curve. An envelope of the one-parameter family of integral curves is a curve in the xy-plane such that at each point it is tangent to one of the integral curves. Since there is no universally accepted definition of a singular solution, some authors define a singular solution as an envelope of the family of integral curves obtained from the general solution. Our definition of a singular solution includes not only the envelopes, but all solutions that have branch points. This broader definition is motivated by practical applications of differential equations in modeling real-world problems. The existence of a singular solution gives a warning signal in using the differential equation as a reliable model.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
A singular solution of special interest is one that consists entirely of branch points-at every point it is tangent to another integral curve. An envelope of the one-parameter family of integral curves is a curve in the xy-plane such that at each point it is tangent to one of the integral curves. Since there is no universally accepted definition of a singular solution, some authors define a singular solution as an envelope of the family of integral curves obtained from the general solution. Our definition of a singular solution includes not only the envelopes, but all solutions that have branch points. This broader definition is motivated by practical applications of differential equations in modeling real-world problems. The existence of a singular solution gives a warning signal in using the differential equation as a reliable model.
Approximate Analytical Methods
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
A singular solution. Recall that “A singular solution of y′=f(x,y) is a function that is not a special case of the general solution and for which the uniqueness of the initial value problem has failed.” Dobrushkin [359, Section 1.4].
Free energy in vertically aligned liquid crystal mode
Published in Liquid Crystals, 2018
Jieh-Wen Tsung, Tien-Lun Ting, Wen-Hao Hsu
kd is directly related to the feature of the defect. In our case, kd is +1 or −1 for the defects shown in Figure 1(b). r is the radial length in cylindrical coordinates, now indicating the distance from the defect core. rc and Fc are the diameter and the free energy of the defect core, respectively. All the physical parameters in Equations (6)–(8) are listed in Table 1. This planar, singular solution shows that felastic increases logarithmically when r approaches to 0, with a singularity at the core where the Oseen–Frank theory no longer holds.
Symmetry analysis, laws of conservation, and numerical and approximate analysis of Burger’s fractional order differential equation
Published in Waves in Random and Complex Media, 2023
Shaban Mohammadi, S. Reza Hejazi
Due to characteristics such as the simplicity of constructing and developing non-standard finite difference schemes, these schemes are employed to numerically solve various differential equations generated in relation to engineering and natural science. Historically, for each ordinary differential equation with the initial condition that it has a singular solution, Mickens proved that there is an exact difference equation with zero local truncation error [2]. Nevertheless, the exact general solution to the differential equation is needed to construct the exact difference equation. Assuming that structure of standard finite difference schemes to solve differential equations may cause non-suitable behavior for the solutions (e.g. numerical instability and disheveled behavior). Mickens presented the following rules for constructing non-standard finite difference schemes: Nonlinear terms in the differential equation should be replaced by discrete localized approximates.To make the first-order derivative in a differential equation discrete, the current denominator of the fraction, namely , can be replaced by more complex negative functions.The order of the discrete derivatives must be exactly equal to the order of the corresponding derivatives of the differential equations.Special solutions of the differential equations should also be special solutions of the non-standard finite difference scheme.The scheme should preserve the qualitative behavior of exact solutions to the problem.To describe non-standard finite difference scheme, we assume the following differential equation: where a parameter and Euler’s discretization method is one of the most straightforward techniques. In this method, the derivative term of is replaced by . However, in the Mickens scheme, this term is replaced by . Next, assume the following discretization: The non-standard finite difference scheme is constructed using the following two steps:The derivative in the left side of Equation (32) is replaced by discretized form: is an approximate of .