Editing Partial differential equation (section)

== Analytical solutions ==

===Separation of variables===
{{main|Separable partial differential equation}}
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is ''the'' solution (this also applies to ODEs). We assume as an [[ansatz]] that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.<ref>{{cite book |last1=Gershenfeld |first1=Neil |title=The nature of mathematical modeling |url=https://archive.org/details/naturemathematic00gers_334 |url-access=limited|date=2000|publisher=Cambridge University Press|location=Cambridge|isbn=0521570956|page=[https://archive.org/details/naturemathematic00gers_334/page/n32 27]|edition=Reprinted (with corr.)}}</ref>

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

This is possible for simple PDEs, which are called [[separable partial differential equation]]s, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to [[diagonal matrices]] – thinking of "the value for fixed {{mvar|x}}" as a coordinate, each coordinate can be understood separately.

This generalizes to the [[method of characteristics]], and is also used in [[integral transform]]s.

===Method of characteristics===
{{main|Method of characteristics}}
The characteristic surface in {{math|1=''n'' = ''2''-}}dimensional space is called a '''characteristic curve'''.{{sfn|Zachmanoglou|Thoe|1986|pp=115–116}} 
In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the [[method of characteristics]].

More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.

===Integral transform===
An [[integral transform]] may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

An important example of this is [[Fourier analysis]], which diagonalizes the heat equation using the [[eigenbasis]] of sinusoidal waves.

If the domain is finite or periodic, an infinite sum of solutions such as a [[Fourier series]] is appropriate, but an integral of solutions such as a [[Fourier integral]] is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

===Change of variables===

Often a PDE can be reduced to a simpler form with a known solution by a suitable [[Change of variables (PDE)|change of variables]]. For example, the [[Black–Scholes equation]]
<math display="block"> \frac{\partial V}{\partial t} + \tfrac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 </math>
is reducible to the [[heat equation]]
<math display="block"> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}</math>
by the change of variables<ref>{{cite book |first1=Paul |last1=Wilmott |first2=Sam |last2=Howison |first3=Jeff |last3=Dewynne |title=The Mathematics of Financial Derivatives |location= |publisher=Cambridge University Press |year=1995 |isbn=0-521-49789-2 |pages=76–81 |url=https://books.google.com/books?id=VYVhnC3fIVEC&pg=PA76 }}</ref>
<math display="block">\begin{align}
V(S,t) &= v(x,\tau),\\[5px]
x &= \ln\left(S \right),\\[5px]
\tau &= \tfrac{1}{2} \sigma^2 (T - t),\\[5px]
v(x,\tau) &= e^{-\alpha x-\beta\tau} u(x,\tau).
\end{align}</math>

===Fundamental solution===
{{main|Fundamental solution}}
Inhomogeneous equations{{clarification needed|date=July 2020}} can often be solved (for constant coefficient PDEs, always be solved) by finding the [[fundamental solution]] (the solution for a point source <math>P(D)u=\delta</math>), then taking the [[convolution]] with the boundary conditions to get the solution.

This is analogous in [[signal processing]] to understanding a filter by its [[impulse response]].

===Superposition principle===
{{further| Superposition principle }}
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example {{math|1=sin ''x'' + sin ''x'' = 2 sin ''x''}}. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} are solutions of linear PDE in some function space {{mvar|R}}, then {{math|1=''u'' = ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}} with any constants {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are also a solution of that PDE in the same function space.

===Methods for non-linear equations===
{{see also|nonlinear partial differential equation}}

There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the [[Cauchy–Kowalevski theorem]]) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of [[mathematical analysis|analysis]]).

Nevertheless, some techniques can be used for several types of equations. The [[h-principle|{{mvar|h}}-principle]] is the most powerful method to solve [[Underdetermined system|underdetermined]] equations. The [[Riquier–Janet theory]] is an effective method for obtaining information about many analytic [[Overdetermined system|overdetermined]] systems.

The [[method of characteristics]] can be used in some very special cases to solve nonlinear partial differential equations.<ref>{{cite book |first=J. David |last=Logan |title=An Introduction to Nonlinear Partial Differential Equations |location=New York |publisher=John Wiley & Sons |year=1994 |isbn=0-471-59916-6 |chapter=First Order Equations and Characteristics |pages=51–79 }}</ref>

In some cases, a PDE can be solved via [[perturbation analysis]] in which the solution is considered to be a correction to an equation with a known solution. Alternatives are [[numerical analysis]] techniques from simple [[finite difference]] schemes to the more mature [[multigrid]] and [[finite element method]]s. Many interesting problems in science and engineering are solved in this way using [[computer]]s, sometimes high performance [[supercomputer]]s.

===Lie group method===
From 1870 [[Sophus Lie]]'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called [[Lie group]]s, be referred, to a common source; and that ordinary differential equations which admit the same [[infinitesimal transformation]]s present comparable difficulties of integration. He also emphasized the subject of [[contact transformation|transformations of contact]].

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous [[infinitesimal transformation]]s of solutions to solutions ([[Lie theory]]). Continuous [[group theory]], [[Lie algebras]] and [[differential geometry]] are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its [[Lax pair]]s, recursion operators, [[Bäcklund transform]] and finally finding exact analytic solutions to the PDE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

===Semi-analytical methods===
The [[Adomian decomposition method]],<ref>{{cite book |title=Solving Frontier problems of Physics: The decomposition method |first=G. |last=Adomian|author-link=George Adomian|publisher=Kluwer Academic Publishers |year=1994 |isbn=9789401582896 |url=https://books.google.com/books?id=UKPqCAAAQBAJ&q=%22partial+differential%22}}</ref> the [[Aleksandr Lyapunov|Lyapunov]] artificial small parameter method, and his [[homotopy perturbation method]] are all special cases of the more general [[homotopy analysis method]].<ref>{{cite book | last=Liao | first=S. J. |author-link=Liao Shijun| title=Beyond Perturbation: Introduction to the Homotopy Analysis Method | publisher=Chapman & Hall/ CRC Press | location=Boca Raton | year=2003 | isbn=1-58488-407-X }}</ref> These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known [[perturbation theory]], thus giving these methods greater flexibility and solution generality.