Μερική Διαφορική Εξίσωσις
In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. They find their generalization in stochastic partial differential equations. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems.
A partial differential equation (PDE) for the function is of the form
F is a linear function of u and its derivatives if, by replacing u with v+w, F can be written as and if, by replacing u with ku, F can be written as
A relatively simple PDE is
This relation implies that the function u(x,y) is independent of x. Hence the general solution of this equation is
where f is an arbitrary function of y. The analogous ordinary differential equation is
which has the solution
where c is any constant value (independent of x). These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function can be determined if is specified on the line .
Existence and uniqueness Edit
Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. There is a general theorem (the Cauchy–Kowalevski theorem) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.
An example of pathological behavior is the sequence of Cauchy problems (depending upon n) for the Laplace equation
with boundary conditions
where n is an integer. The derivative of u with respect to y approaches 0 uniformly in x as n increases, but the solution is
This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called ill-posed or not well posed, since the solution does not depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.
In PDEs, it is common to denote partial derivatives using subscripts.
Especially in (mathematical) physics, one often prefers the use of del (which in cartesian coordinates is written ) for spatial derivatives and a dot for time derivatives. For example, the wave equation (described below) can be written as
- (physics notation),
- (math notation), where Δ is the Laplace operator.
This often leads to misunderstandings in regards of the -(delta)operator.
Heat equation in one space dimension Edit
The equation for conduction of heat in one dimension for a homogeneous body has the form
where u(t,x) is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying , where f(x) is an arbitrary function.
General solutions of the heat equation can be found by the method of separation of variables. Some examples appear in the heat equation article. They are examples of Fourier series for periodic f and Fourier transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the form
where F is an arbitrary function. To satisfy the initial condition, F is given by the Fourier transform of f, that is
If f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source. For a source whose strength is normalized to 1, the result is
and the resulting solution of the heat equation is
This is a Gaussian integral. It may be evaluated to obtain
This result corresponds to a normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena.
Wave equation in one spatial dimension Edit
The wave equation is an equation for an unknown function u(t, x) of the form
Here u might describe the displacement of a stretched string from equilibrium, or the difference in air pressure in a tube, or the magnitude of an electromagnetic field in a tube, and c is a number that corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of a string or other medium:
where f and g are arbitrary given functions. The solution of this problem is given by d'Alembert's formula:
This formula implies that the solution at (t,x) depends only upon the data on the segment of the initial line that is cut out by the characteristic curves
that are drawn backwards from that point. These curves correspond to signals that propagate with velocity c forward and backward. Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves. This behavior is very different from the solution for the heat equation, where the effect of a point source appears (with small amplitude) instantaneously at every point in space. The solution given above is also valid if t is negative, and the explicit formula shows that the solution depends smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-posed.
Spherical waves Edit
Spherical waves are waves whose amplitude depends only upon the radial distance r from a central point source. For such waves, the three-dimensional wave equation takes the form
This is equivalent to
and hence the quantity ru satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form
where F and G are completely arbitrary functions. Radiation from an antenna corresponds to the case where G is identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1/r. This feature of undistorted propagation of waves is not present if there are two spatial dimensions.
Laplace equation in two dimensions Edit
The Laplace equation for an unknown function of two variables φ has the form
Solutions of Laplace's equation are called harmonic functions.
Connection with holomorphic functions Edit
Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy–Riemann equations state that
and it follows that
Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally. Details are given in Laplace equation.
A typical boundary value problem Edit
A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain. For example, we may seek a harmonic function that takes on the values u(θ) on a circle of radius one. The solution was given by Poisson:
Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If r<1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable. This behavior is typical for solutions of elliptic partial differential equations: the solutions may be much more smooth than the boundary data. This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data.
Euler–Tricomi equation Edit
Advection equation Edit
The advection equation describes the transport of a conserved scalar in a velocity field . It is:
If the velocity field is solenoidal (that is, ), then the equation may be simplified to
In the one-dimensional case where is not constant and is equal to , the equation is referred to as Burgers' equation.
Ginzburg–Landau equation Edit
where and are constants and is the imaginary unit.
The Dym equation Edit
Initial-boundary value problems Edit
Many problems of mathematical physics are formulated as initial-boundary value problems.
Vibrating string Edit
If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0<x<L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions
as well as the initial conditions
The method of separation of variables for the wave equation
leads to solutions of the form
where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form
where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n=1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes. Wind instruments typically correspond to vibrations of an air column with one end open and one end closed. The corresponding boundary conditions are
The method of separation of variables can also be applied in this case, and it leads to a series of odd overtones.
The general problem of this type is solved in Sturm–Liouville theory.
Vibrating membrane Edit
If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation
if t>0 and (x,y) is in D. The boundary condition is if is on . The method of separation of variables leads to the form
which in turn must satisfy
The latter equation is called the Helmholtz Equation. The constant k must be determined to allow a non-trivial v to satisfy the boundary condition on C. Such values of k2 are called the eigenvalues of the Laplacian in D, and the associated solutions are the eigenfunctions of the Laplacian in D. The Sturm–Liouville theory may be extended to this elliptic eigenvalue problem (Jost, 2002).
Other examples Edit
Except for the Dym equation and the Ginzburg–Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier–Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.
Also see the list of non-linear partial differential equations.
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or elliptic. Others such as the Euler–Tricomi equation have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions, and to smoothness of the solutions.
Equations of first order Edit
Equations of second order Edit
Assuming , the general second-order PDE in two independent variables has the form
where the coefficients A, B, C etc. may depend upon x and y. If over a region of the xy plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
More precisely, replacing by and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the top degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is with the factor of 4 dropped for simplicity.
- : solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x<0.
- : equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x=0.
- : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x>0.
If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form
The classification depends upon the signature of the eigenvalues of the coefficient matrix.
- Elliptic: The eigenvalues are all positive or all negative.
- Parabolic : The eigenvalues are all positive or all negative, save one that is zero.
- Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
- Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
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 are m by m matrices for . The partial differential equation takes the form
where the coefficient matrices Aν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form
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:
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 space-like 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
has m real roots λ1, λ2, ..., λm. 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 = λ ξ 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.
Equations of mixed type Edit
If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler–Tricomi equation
which is called elliptic-hyperbolic because it is elliptic in the region x < 0, hyperbolic in the region x > 0, and degenerate parabolic on the line x = 0.
Infinite-order PDEs in quantum mechanics Edit
Weyl quantization in phase space leads to quantum Hamilton's equations for trajectories of quantum particles. Those equations are infinite-order PDEs. However, in the semiclassical expansion one has a finite system of ODEs at any fixed order of . The equation of evolution of the Wigner function is infinite-order PDE also. The quantum trajectories are quantum characteristics with the use of which one can calculate the evolution of the Wigner function.
Analytical methods to solve PDEs Edit
Separation of variablesEdit
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.
Method of characteristicsEdit
Πρότυπο:Main 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.
An integral transform may transform the PDE to a simpler one, in particular a separable PDE. This corresponds to diagonalizing an operator.
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 for use of a Fourier integral.
Change of variablesEdit
is reducible to the heat equation
by the change of variables (for complete details see Solution of the Black Scholes Equation)
Πρότυπο:Main Inhomogeneous equations 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.
Because any superposition of solutions of a linear, homogeneous PDE is again a solution, the particular solutions may then be combined to obtain more general solutions.
Methods for non-linear equationsEdit
- See also the list of nonlinear partial differential equations.
There are no generally applicable methods to solve non-linear 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.
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 MethodsEdit
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 solve PDE's 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.
Numerical methods to solve PDEs Edit
The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM). The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other versions of FEM include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), etc.
Finite Element Method Edit
Πρότυπο:Main The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc.
Finite Difference Method Edit
Πρότυπο:Main Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Finite Volume Method Edit
Πρότυπο:Main Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.
Almost-solution of PDEEdit
- Φυσικός Νόμος
- Boundary value problem
- Difference equation
- Laplace transform applied to differential equations
- List of dynamical systems and differential equations topics
- Matrix differential equation
- Ordinary differential equation
- Separation of variables
- Stochastic partial differential equations
- Numerical partial differential equations
- Stochastic processes and boundary value problems
- Dirichlet boundary condition
- Neumann boundary condition
- Robin boundary condition
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