# What are some examples of parabolic PDEs

## Partial differential equations

Transcript

1 Partial differential equations General remarks Author: Harald Höller last change: license: Creative Commons license by-nc-sa 3.0 at Definition of a PDE A partial differential equation (PDE) is an equation in which a function (which depends on at least two variables) and their derivatives occur according to at least two variables. If one represents such an equation implicitly, then that of the form F x, y, ux, y, xux, y, yux, y, x, yux, y, ... 0 More general for a system of equations (F) and any dimension of the unknown function u (x) and differential operator D. F x, ux, D ux, D 2 ux, ..., D kux 0 A differential equation is always completed by specifying (suitable) boundary resp. Initial conditions. A PDE + RB is called correctly posed (well posed) if (at least) one solution exists the solution is unambiguous the solution constantly depends on the RB (AB) In general, non-differentiable (weak) solutions are also allowed. Classification The most obvious properties of differential equations are their dimension (number of independent variables) and their order (the degree of the highest occurring derivative). Further criteria are the degree of inearity (linear in all derivatives, linear in the highest derivative, etc.). The most important classification for the specification of boundary conditions, the solution path and the properties of the solution of the equation itself is based on the coefficients of the derivative terms. For a second order PDE in two dimensions we consider the expression a x, y x, x u x, y b x, y x, y u x, y c x, y y, y u x, y d x, y x u x, y e x, y y u x, y f u x, y, x, y 0

2 2 PDEs_m6.nb Based on the conic sections, the PDE is called elliptical if a (x, y) c (x, y) - bx, y 2/4> 0 parabolic if a (x, y) c (x, y) - bx, y 2/4 = 0 hyperbolic if a (x, y) c (x, y) - bx, y 2/4 <0 Since the coefficients in the general (nonlinear) case themselves depend on the location, so is the type the PDE depending on the location. Properties and boundary conditions The types of PDEs defined above have very different properties, which are briefly presented below. For the sake of simplicity, equations are considered that can be assigned to a type globally, i.e. I limit myself to constant coefficients. In physics, PDEs usually appear in 3 space and one time dimensions, so the terms "stationary" and "unsteady" in the following refer to the time variable. Elliptical PDEs Elliptical PDEs mostly occur in connection with time-independent (stationary) problems; The most important examples of elliptical problems are the Aplace equation u 0 and the Poisson equation u Ρ These equations describe e.g. the static temperature distribution in a body, the potential of a charge distribution or the gravitational potential of a mass density distribution. The solutions to these equations are stationary fields (u), i.e. information in this system is prepared "instantaneously". The boundary conditions for elliptic equations are divided into Dirichlet's, Neumann's and mixed equations. In the case of Dirichtlet's boundary condition, values of u are given at the boundary, i.e. e.g. a fixed temperature at the edge of a body, realized by a heat bath (u = const). Neumann's boundary conditions set the normal derivative at the edge of the area, i.e. for an isolated body the change in temperature at the edge can be set to zero (xu 0. Two continuously differentiable solutions of the Aplace equation are called harmonic functions, for which a maximum principle is formulated To put it loosely, it means that a harmonic function in a restricted area assumes its maximum (or minimum) at the edge (note: this is a very strong statement). Another important property of elliptical equations is the regularizing property, that of the DOP (with this sign). If one gives arbitrarily discontinuous values at the edge for an elliptic equation, then the solution inside the area is always analytic. Furthermore, it can be shown that if the source function Ρ in the Poisson equation is continuous, is u already continuously differentiable twice in the whole area. We will see later with the hyperbolic PDEs that these statements are by no means true there.

3 PDEs_m6.nb As an example from Wolfram Demonstrations Project: the aplace equation on a circle with boundary condition u = sin (kθ), where k can be manipulated. Contributed by: David von Seggern (University Nevada-Reno) Manipulate solution is specific to BC Sin k Θ. ParametricPlot3D r Cos Θ, r Sin Θ, r ^ k Sin k Θ, r, 0, 1, Θ, 0, 2 Π, PlotRange 1, 1, PlotPoints 26, 18 k 1, BoxRatios 1, 1, 1, ViewPoint 10 , 5, 5, Axesabel Style "x", Italic, Style "y", Italic, Style "U", Italic, ImageSize 450, 450, k, 1, "number of cycles", 1, 12, 1, Appearance " abeled "number of cycles 5

4 4 PDEs_m6.nb Parabolic PDEs Parabolic PDEs describe similar phenomena and have similar properties to elliptical PDEs, but for the unsteady case. The most important example is the heat conduction equation Σ u t u 0 which e.g. describes the development over time (cooling, heating) of a temperature distribution. The propagation of the "information" in this case is essentially determined by the coefficient of thermal conductivity Σ, which corresponds to a (not necessarily constant) material constant. As above, Dirichlet, Neumann or mixed RB occur as local boundary conditions, and an initial condition must also be specified. The so-called characteristics of the heat conduction equation are the families {t = const} on which a condition has to be given so that the system can be solved. In the case of the heat conduction equation, too, the aplace operator shows its regularizing side. Any discontinuous initial values (anywhere in the area) lead to analytical solutions within a very short time (in terms of time development). In physical terms, this means that gradients are broken down very quickly and an equilibrium is established over time under any initial conditions; the initial data are to a certain extent "forgotten".

5 PDEs_m6.nb Hyperbolic PDEs Hyperbolic PDEs describe phenomena with finite propagation speed; Examples are the wave equation Σ u t, t u c 2 0 or the advection equation. Note: all (simple) spatial derivatives x are to be understood as Nabla operators in the following; the functions they act on are not explicitly marked as vector-valued (results from the context)! v x u t u 0 Further examples are the spin field equations (Weyl, Dirac), the Maxwell equations, basic equations of hydrodynamics, etc. Hyperbolic initial value problems (Cauchy problems) are determined by initial data on the characteristics. The characteristics of the wave equation in 1 + 1 dimensions are the forward and backward directed cones of light {(t, x) x- x 0 = + - c (t- t 0}. Hyperbolic PDEs describe causal fields, which is due to the mentioned finite speed of propagation In the case of the wave equation, this propagation speed is, for example, the speed of sound or the speed of light. Outside the cone of sound or light, no information is transmitted. In contrast to the two types of PDEs described above, the solutions of hyperbolic equations are only very weak or even very weak not damped. Wavy phenomena can even split up in such a way that discontinuities (shocks) occur. The solution theory of hyperbolic PDEs is therefore a lot more difficult because less differentiable functions occur Wiki of the Faculty of Physics in the respective work area rich from. 1: "Mathematica solution" from M2 Ex. 13) Determine the temperature distribution of a thin wire with initial and boundary conditions as in M2 Example 13) Thermally conductive wire specified numerically with Mathematica. You will need two commands: NDSolve and Derivative. Also work out the following points: 1.1: How do Neumann's boundary conditions have to be defined in Mathematica and why? 1.2: In what form does Mathematica output the solution and what does that mean for your "result"?

6 6 PDEs_m6.nb? NDSolve NDSolve eqns, y, x, x min, x max finds a numerical solution to the ordinary differential equations eqns for the function y with the independent variable x in the range x min to x max.NDSolve eqns, y, x, x min , x max, t, t min, t max finds a numerical solution to the partial differential equations eqns. NDSolve eqns, y 1, y 2 ,, x, x min, x max finds numerical solutions for the functions y i.? Derivative f 'represents the derivative of a function f of one argument. Derivative n 1, n 2, f is the general form, representing a function obtained from f by differentiating n 1 times with respect to the first argument, n 2 times with respect to the second argument, and so on. DSolve DT x, t, t DT x, t, x, 2, Derivative 1, 0 T 0, t 0, Derivative 1, 0 T Pi, t 0, T x, 0 6 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, T x, t, x, t DSolve T 0.1 x, t T 2.0 x, t, T 1.0 0, t 0, T 1.0 Π, t 0, T x, 0 6 Π x2 2 Π 3 x3 3, T x, t, x, t NDSolve DT x, t, t DT x, t, x, 2, Derivative 1, 0 T 0, t 0, Derivative 1, 0 T Pi, t 0, T x, 0 6 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, T, x, 0, Pi, t, 0, 5 T InterpolatingFunction 0.,, 0., 5., 2: Graphical representation and discussion of the solution Plot the solution with the command Plot3D or ContourPlot (and PlotRange) and discuss the timing behavior. Export the graphics from Mathematica and post them in the form of images on the Wiki page. Also discuss the following questions: 2.1: How does the solution change if the boundary conditions are not Neumann's but Dirichlet's. Calculate the example for such a case by plot and discuss the solution.

7 PDEs_m6.nb Plot3D T x, t., X, 0, Pi, t, 0, 3, PlotRange All NDSolve DT x, t, t DT x, t, x, 2, T 0, t Pi 2 x ^ 2 x ^ 3 3, T Pi, t 6 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, T x, 0 6 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, T, x, 0, Pi, t , 0, 5 T InterpolatingFunction 0.,, 0., 5., Plot3D T x, t., X, 0, Pi, t, 0, 3, PlotRange All

8 8 PDEs_m6.nb 3: Comparison with the analytical approximate solution from M2 Ex. 13) Compare the approximate solution proposed in M2 Ex. 13c) graphically with the numerical solution from Mathematica. 3.1: How does the approximate solution change the more terms you consider in the Fourier expansion? 3.2: How could one determine the difference between the approximate solution and the (analytically determined) Fourier series besides graphically? DSolve k ^ 2 g '' x g x Λ, g '0 0, g' 0, g, x g Function x, 0? Reduce Reduce expr, vars reduces the statement expr by solving equations or inequalities for vars and eliminating quantifiers. Reduce expr, vars, dom does the reduction over the domain dom. Common choices of domains are reals, integers and complexes. Reduce Refine Sin l 0, l, assumptions Λ Integers, k Reals k C 1 Integers && k 0 && 0 0 && k 0 && l 2 k Π C 1 lk Π 2 Π C 1 DSolve f 'tft Λ, f 0 6 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, f, tf Function t, t Λ 3 Π 2 xx 2 Π 3 gana x_, k_ C 1 Cos xk Λ. Λ 2 k Π n. C 1 1 Cos 2 n Π x Sum gana x, n, n, 0, 2 1 Cos 2 Π x 4 Π x Cos

9 PDEs_m6.nb Needs "FourierSeries`"? FourierSeries FourierSeries expr, t, n gives the order n Fourier exponential series expansion of expr, where expr is a periodic function of t with period 1. FourierSeries 1 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, x, 1 FullSimplify Expand 1 Cos 2 Π x 24 Π2 2 Π 4 Sin 2 Π x Sin 2 Π x 2 Π 6 12 Π 4 Plot 1 Pi ^ 3 Pi 2 x ^ 2 x ^ 3 3, 1 Cos 2 Π x 24 Π2 2 Π 4 Sin 2 Π x Sin 2 Π x, x, 0, Pi 2 Π 6 12 Π Solve DC 1 Cos xk Λ C 2 Sin xk Λ, x 0 Sum Cos 2 Pi nx, Sin 2 Π x

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