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Initial value problem. In multivariable calculus, an initial value problem [a] ( IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem.
Laplace's method. In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form. where is a twice- differentiable function, M is a large number, and the endpoints a and b could possibly be infinite. This technique was originally presented in Laplace (1774) .
Laplace transform. In mathematics, the Laplace transform, named after Pierre-Simon Laplace ( / ləˈplɑːs / ), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane ).
List of Laplace transforms. The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency ).
Laplace's equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known as potential theory . The twice continuously differentiable solutions of Laplace's equation are the harmonic functions , [1] which are important in multiple branches of physics, notably electrostatics, gravitation ...
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions . First consider the following property of the Laplace transform:
Laplace expansion. In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n - matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) - submatrices of B. Specifically, for every i, the Laplace expansion ...
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian - or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.