Systems of first-order equations

Systems of first-order equations and characteristic surfaces[edit]

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown $u$ is now a vector with $m$ components, and the coefficient matrices $A ν$ are $m$ by $m$ matrices for $ν = 1, 2, \dots, n$ . The partial differential equation takes the form

Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,

where the coefficient matrices

A ν

and the vector

B

may depend upon

x

and

u

. If a hypersurface

S

is given in the implicit form

\varphi (x_{1},x_{2},\ldots ,x_{n})=0,

where

φ

has a non-zero gradient, then

S

is a characteristic surface for the operator

L

at a given point if the characteristic form vanishes:

Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.

The geometric interpretation of this condition is as follows: if data for $u$ are prescribed on the surface $S$ , then it may be possible to determine the normal derivative of $u$ on $S$ from the differential equation. If the data on $S$ and the differential equation determine the normal derivative of $u$ on $S$ , then $S$ is non-characteristic. If the data on $S$ and the differential equation do not determine the normal derivative of $u$ on $S$ , then the surface is characteristic, and the differential equation restricts the data on $S$ : the differential equation is internal to $S$ .

A first-order system $Lu = 0$ is elliptic if no surface is characteristic for $L$ : the values of $u$ on $S$ and the differential equation always determine the normal derivative of $u$ on $S$ .
A first-order system is hyperbolic at a point if there is a spacelike surface $S$ with normal $ξ$ at that point. This means that, given any non-trivial vector $η$ orthogonal to $ξ$ , and a scalar multiplier $λ$ , the equation $Q (λξ + η) = 0$ has $m$ real roots $λ 1, λ 2, \dots, λ m$ . The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form $Q (ζ) = 0$ defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has $m$ sheets, and the axis $ζ = λξ$ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

Analytical solutions[edit]

Separation of variables[edit]

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic 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.^[5]

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 equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed $x$ " as a coordinate, each coordinate can be understood separately.

This generalizes to the method of characteristics, and is also used in integral transforms.

Method of characteristics[edit]

In special cases, one can find characteristic curves on which the equation 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, one may find characteristic surfaces. For a second order partial differential equation solution, see the Charpit method.

Integral transform[edit]

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[edit]

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes equation

{\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

is reducible to the heat equation

{\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}

by the change of variables^[6]

{\begin{aligned}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{aligned}}

Fundamental solution[edit]

Inhomogeneous equations^{[clarification needed]} can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), 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[edit]

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 $sin x + sin x = 2 sin x$ . The same principle can be observed in PDEs where the solutions may be real or complex and additive. If $u 1$ and $u 2$ are solutions of linear PDE in some function space $R$ , then $u = c 1 u 1 + c 2 u 2$ with any constants $c 1$ and $c 2$ are also a solution of that PDE in the same function space.

Methods for non-linear equations[edit]

There are no generally applicable 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 analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

Nevertheless, some techniques can be used for several types of equations. The $h$ -principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations.^[7]

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 methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

Lie group method[edit]

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 groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations 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 pairs, 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.

Semianalytical methods[edit]

The Adomian decomposition method,^[8] the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method.^[9] 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.

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Dr. Balamurugan