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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 ).
The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.
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:
To counteract this problem, classical control theory uses the Laplace transform to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the frequency domain. Once a given system has been converted into the frequency domain it can be manipulated with greater ease.
Consider the following second-order problem, ′ + + = () =, where = {,, <is the Heaviside step function.The Laplace transform is defined by, = {()} = ().Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,
Two-sided Laplace transform; Inverse two-sided Laplace transform; Laplace–Carson transform; Laplace–Stieltjes transform; Legendre transform; Linear canonical transform; Mellin transform. Inverse Mellin transform; Poisson–Mellin–Newton cycle; N-transform; Radon transform; Stieltjes transformation; Sumudu transform; Wavelet transform ...
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by = + + (), with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by using either the classical method of solving linear differential equations or the Laplace transform method.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...