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The Dominant Balance and WKB Methods
Published in Alan W. Bush, Perturbation Methods for Engineers and Scientists, 2018
The Frobenius method allows the solution of the differential equation near the point x = x0 to be represented as an infinite series in powers of (x – x0). The condition which must be satisfied for the Frobenius method to be successful is that the point x = x0 be either an ordinary point of the equation when p(x0) and q(x0) are finite, or a regular singular point when Limx→x0[(x−x0)p(x)]andLimx→x0[(x−x0)2q(x)]
Power series methods of solving ordinary differential equations
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
A differential equation of the form y′′+Py′+Qy=0, where P and Q are both functions of x, such that the equation can be represented by a power series, may be solved by the Frobenius method.3
Differential Equations for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The Frobenius method is used to second-order differential equations with non-constant coefficients of the form: x2y″+P(x)xy′+Q(x)y =0
Dynamic response of an axisymmetric transversely isotropic medium with its modulus varying with depth subjected to LWD load
Published in International Journal of Pavement Engineering, 2022
Haishan Fan, Junhui Zhang, Shiping Zhang
Furthermore, by substituting Equation (20) into Equations (21)–(22), the following expression can be obtained. For the case of Ev0 < Ev∞For the case of Ev0 > Ev∞where δ = Ev0/Ev∞. Equations (26)–(29) are the systems of ordinary differential equations with variable coefficients with respect to ψ. Mathematically, the Frobenius method is suitable to solve the above problems, and the solutions are usually assumed to be in a power series form. In this paper, the solution of the and can be assumed to be the form with power series: where m is an undetermined constant, which is related to the boundary conditions. By the substitution of displacement components in Equations (26)–(29) by their expressions in Equation (30), the following expressions can be obtained.