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The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems. In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.
The homotopy analysis method ( HAM) is a semi-analytical technique to solve nonlinear ordinary / partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy- Maclaurin series to deal with the ...
The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration. [1] It was one of the first advanced computing devices to be used operationally. [2] The original machines could not add, but then it was noticed that if the two wheels ...
The same illustration for The midpoint method converges faster than the Euler method, as . Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to ...
In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for ...
The question of finding solutions to such equations is known as the Dirichlet problem. In the sciences and engineering, a Dirichlet boundary condition may also be referred to as a fixed boundary condition or boundary condition of the first type. It is named after Peter Gustav Lejeune Dirichlet (1805–1859).
Summary of exact solutions. Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here. In the table below, P(x), Q(x), P(y), Q(y), and M(x,y), N(x,y) are any integrable functions of x, y; b and c are real given constants; C 1, C 2, ... are arbitrary constants (complex in ...
Differential equations. In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form. where is a real number. Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [3] [4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named.