In mathematics, the

Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...

is a powerful

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...

used to switch a function from the

time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...

to the

s-domain In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...

. The Laplace transform can be used in some cases to solve

linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...

s with given

initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...

. First consider the following property of the Laplace transform: :

\mathcal\=s\mathcal\-f(0)

\mathcal\=s^2\mathcal\-sf(0)-f'(0)

One can prove by induction that :

\mathcal\=s^n\mathcal\-\sum_^s^f^(0)

Now we consider the following differential equation: :

\sum_^a_if^(t)=\phi(t)

with given initial conditions :

f^(0)=c_i

Using the

linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

of the Laplace transform it is equivalent to rewrite the equation as :

\sum_^a_i\mathcal\=\mathcal\

obtaining :

\mathcal\\sum_^a_is^i-\sum_^\sum_^a_is^f^(0)=\mathcal\

Solving the equation for

\mathcal\

and substituting

f^(0)

with

c_i

one obtains :

\mathcal\=\frac

The solution for ''f''(''t'') is obtained by applying the

inverse Laplace transform In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the ...

\mathcal\.

Note that if the initial conditions are all zero, i.e. :

f^(0)=c_i=0\quad\forall i\in\

then the formula simplifies to :

f(t)=\mathcal^\left\

An example

We want to solve :

f''(t)+4f(t)=\sin(2t)

with initial conditions ''f''(0) = 0 and ''f′''(0)=0. We note that :

\phi(t)=\sin(2t)

and we get :

\mathcal\=\frac

The equation is then equivalent to :

s^2\mathcal\-sf(0)-f'(0)+4\mathcal\=\mathcal\

We deduce :

\mathcal\=\frac

Now we apply the Laplace inverse transform to get :

f(t)=\frac\sin(2t)-\frac\cos(2t)

Bibliography

* A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. {{isbn, 1-58488-299-9 Integral transforms Differential equations Differential calculus Ordinary differential equations