Parabolic partial differential equation

Parabolic partial differential equations are a special class of second or higher order partial differential equations (PDG) that are used in describing a wide range of scientific problems. These are so-called evolution problems in which a “time variable” appears and the development in “time” is described using a first-order derivation. The solutions of parabolic differential equations often behave like the solutions of the heat conduction equation , which describes the heat conduction in solids or the diffusion in liquids and gases.

Generalizing the heat conduction equation, one obtains the important class of linear parabolic second order PDG. In addition to thermal conduction, these are also used, for example, to calculate the propagation of sound in the sea or to develop stock options ( Black-Scholes model ). In the following, only parabolic differential equations of the second order are considered.

Definition in the linear case

Two dimensions

The general linear partial differential equation of the 2nd order with two variables

${\displaystyle a(x,y){\frac {\partial ^{2}u(x,y)}{\partial x^{2}}}+b(x,y){\frac {\partial ^{2}u(x,y)}{\partial x\partial y}}+c(x,y){\frac {\partial ^{2}u(x,y)}{\partial y^{2}}}+d(x,y){\frac {\partial u(x,y)}{\partial x}}+e(x,y){\frac {\partial u(x,y)}{\partial y}}+f(x,y)u(x,y)=0}$

is called parabolic at the point if the coefficient functions of the highest derivatives at the point meet the condition ${\displaystyle (x,y)}$ ${\displaystyle (x,y)}$

${\displaystyle a(x,y)c(x,y)-\left({\frac {b(x,y)}{2}}\right)^{2}=0}$

fulfill. This means that the determinant of the coefficient matrix

${\displaystyle {\begin{pmatrix}a(x,y)&{\frac {b(x,y)}{2}}\\{\frac {b(x,y)}{2}}&c(x,y)\end{pmatrix}}}$

assumes the value 0 at the point . The origin of the term parabolic comes from the analogy of the above coefficient condition to the general conic section equation ${\displaystyle (x,y)}$

${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}$.

If this applies to this equation , then the equation represents a parabola . Similar classifications exist for elliptical and hyperbolic differential equations . ${\displaystyle B^{2}-4AC=0}$

n dimensions

A generalization to several variables is the second order linear partial differential equation

${\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}({\vec {x}}){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+F(t,{\vec {x}},u,\nabla u)=0}$.

As a generalization of the two-dimensional case, the differential equation is called parabolic at the point if the coefficient matrix is positive, semidefinite and singular. This means that all eigenvalues ​​of the coefficient matrix are nonnegative and one eigenvalue vanishes. ${\displaystyle {\vec {x}}}$ ${\displaystyle (a_{i,j}({\vec {x}}))_{i,j}}$

Time-dependent notation

In the last section, the abstract classification was explained as a parabolic differential equation. In many applications the singular direction of the coefficient matrix has the meaning of time . Then the solution is a function that depends on time and position variables . Since the type classification only depends on the coefficients of the highest derivatives, one can also simply allow non-linear dependencies for the lower derivatives. Using coefficient functions and a function represents the equation ${\displaystyle t}$ ${\displaystyle u(t,{\vec {x}})}$ ${\displaystyle t}$ ${\displaystyle n}$ ${\displaystyle {\vec {x}}\in \mathbb {R} ^{n}}$ ${\displaystyle a_{ij}(t,{\vec {x}})}$ ${\displaystyle F}$

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}=&\sum _{i,j=1}^{n}{\dfrac {\partial }{\partial x_{i}}}a_{i,j}(t,{\vec {x}}){\frac {\partial u}{\partial x_{j}}}+F(t,{\vec {x}},u,\nabla u)\\=&\nabla \cdot (a(t,{\vec {x}})\nabla u(t,{\vec {x}}))+F(t,{\vec {x}},u,\nabla u)\end{aligned}}}$

represents a semilinear parabolic differential equation if the matrix of coefficients is positive definite everywhere . The above form of the parabolic differential equation is called the divergence form (based on the divergence operator ). In non-divergence form, a parabolic differential operator is given by ${\displaystyle (a_{i,j}(t,{\vec {x}})_{i,j}}$

${\displaystyle {\frac {\partial u}{\partial t}}=\sum _{i,j=1}^{n}{\widetilde {a}}_{i,j}(t,{\vec {x}}){\frac {\partial ^{2}u}{\partial x_{j}\partial x_{i}}}+F(t,{\vec {x}},u,\nabla u)}$

noted, whereby positive is definite everywhere. The right of the parabolic differential equation is an elliptic differential operator . ${\displaystyle {\widetilde {a}}_{i,j}(t,{\vec {x}})}$ ${\displaystyle \textstyle \sum _{i,j=1}^{n}{\widetilde {a}}_{i,j}(t,{\vec {x}}){\frac {\partial ^{2}}{\partial x_{j}\partial x_{i}}}+F(t,{\vec {x}},\cdot ,\nabla \cdot )}$

Examples

Important examples and classes of parabolic differential equations are

• the heat conduction equation ,
• Reaction-diffusion equations in which the function does not depend on the gradient and the ${\displaystyle F}$ ${\displaystyle \nabla u}$
• Convection-diffusion equations such as the compressible Navier-Stokes equations or the Fokker-Planck equation .

Thermal equation

→ Main article : Heat conduction equation

The most important representative of the linear parabolic differential equation is the heat conduction equation. In one dimension of space it is

${\displaystyle {\frac {\partial }{\partial t}}u(x,t)=a{\frac {\partial ^{2}}{\partial x^{2}}}{u(x,t)}}$.

Here the temperature at the location is at the time , the constant denotes the thermal diffusivity . In several dimensions the equation is through ${\displaystyle u(x,t)}$ ${\displaystyle x}$ ${\displaystyle t}$ ${\displaystyle a}$

${\displaystyle {\frac {\partial }{\partial t}}u(x,t)=a\Delta u(x,t)}$

written down. If one substitutes the parabolic differential equation for the unit matrix in the divergence form , then the main part of this equation is precisely the Laplace operator . This is the standard example of an elliptic differential operator. ${\displaystyle (a_{i,j}(t,{\vec {x}}))_{i,j}=I}$ ${\displaystyle \nabla \cdot (a(t,{\vec {x}})\nabla u(t,{\vec {x}}))=\Delta }$

Initial and marginal values

In most cases, parabolic differential equations are viewed as a combined initial and boundary value problem according to their structure in “space” and “time” variables . If the solution is sought in the interior of a spatial area for times , the initial values ​​at the time are given by a function ${\displaystyle u(x,t)}$ ${\displaystyle G}$ ${\displaystyle t>0}$ ${\displaystyle t=0}$ ${\displaystyle g}$

${\displaystyle u(x,0)=g(x)\quad \forall x\in G}$

before, the boundary values on the boundary of the spatial area are determined for times by a function (or its first spatial derivative) ${\displaystyle \partial G}$ ${\displaystyle G}$ ${\displaystyle t>0}$ ${\displaystyle h}$

${\displaystyle u(x,t)=h(x,t)\quad \forall x\in \partial G,t>0}$

given. Overall, one obtains the initial and boundary value problem

${\displaystyle {\begin{cases}{\frac {\partial u}{\partial t}}(x,t)+Lu(x,t)=f(x,t)&{\text{in}}\ G\times ]0,T]\\u(x,t)=h(x,t)&{\text{auf}}\ \partial G\times [0,T]\\u(x,0)=g(x)&{\text{auf}}\ G\times \{0\}\end{cases}}\,,}$

where is the spatial part of the parabolic differential operator. ${\displaystyle \textstyle L=\sum _{i,j=1}^{n}{\widetilde {a}}_{i,j}(t,{\vec {x}}){\frac {\partial ^{2}}{\partial x_{j}\partial x_{i}}}+\sum _{i=1}^{n}b_{i}(t,{\vec {x}}){\frac {\partial }{\partial x_{i}}}+c(t,{\vec {x}})}$

Harnack inequality

→ Main article : Harnack inequality

Let be the spatial part of the parabolic differential operator and a classical solution of the parabolic differential equation ${\displaystyle \textstyle L=\sum _{i,j=1}^{n}{\widetilde {a}}_{i,j}(t,{\vec {x}}){\frac {\partial ^{2}}{\partial x_{j}\partial x_{i}}}+\sum _{i=1}^{n}b_{i}(t,{\vec {x}}){\frac {\partial }{\partial x_{i}}}+c(t,{\vec {x}})}$ ${\displaystyle u}$

${\displaystyle {\frac {\partial u}{\partial t}}(x,t)+Lu(x,t)=f(x,t)\ {\text{in}}\ G\times ]0,T]}$

with in . Also be a true connected subset. Then exists for a constant such that ${\displaystyle u\geq 0}$ ${\displaystyle G\times ]0,T]}$ ${\displaystyle V\subset \subset G}$ ${\displaystyle 0<t_{1}<t_{2}\leq T}$ ${\displaystyle C}$

${\displaystyle \sup _{V}u(\cdot ,t_{1})\leq C\inf _{V}u(\cdot ,t_{2})}$

applies. The constant is dependent on , , and the coefficient of . ${\displaystyle V}$ ${\displaystyle t_{1}}$ ${\displaystyle t_{2}}$ ${\displaystyle L}$

Maximum principle

→ Main article : Strong maximum principle

Be also the spatial part of the parabolic differential operator, the function is not negative in so and is a classical solution of parabolic differential equation ${\displaystyle \textstyle L=\sum _{i,j=1}^{n}{\widetilde {a}}_{i,j}(t,{\vec {x}}){\frac {\partial ^{2}}{\partial x_{j}\partial x_{i}}}+\sum _{i=1}^{n}b_{i}(t,{\vec {x}}){\frac {\partial }{\partial x_{i}}}+c(t,{\vec {x}})}$ ${\displaystyle c}$ ${\displaystyle G\times ]0,T]}$ ${\displaystyle u\geq 0}$ ${\displaystyle u}$

${\displaystyle {\frac {\partial u}{\partial t}}(x,t)+Lu(x,t)=f(x,t)\ {\text{in}}\ G\times ]0,T]}$

It is also contiguous. ${\displaystyle G}$

• If

${\displaystyle {\frac {\partial }{\partial t}}u(x,t)+Lu(x,t)\leq 0\ {\text{in}}\ G\times ]0,T]}$

holds and assumes a non-negative maximum over in the point , then in is constant . ${\displaystyle u}$ ${\displaystyle {\overline {G}}\times ]0,T]}$ ${\displaystyle (x_{0},t_{0})\in G\times ]0,T]}$ ${\displaystyle u}$ ${\displaystyle G\times ]0,t_{0}]}$

• Analog if

${\displaystyle {\frac {\partial }{\partial t}}u(x,t)+Lu(x,t)\geq 0\ {\text{in}}\ G\times ]0,T]}$

holds and assumes a non-positive maximum over in the point , then in is constant . ${\displaystyle u}$ ${\displaystyle {\overline {G}}\times ]0,T]}$ ${\displaystyle (x_{0},t_{0})\in G\times ]0,T]}$ ${\displaystyle u}$ ${\displaystyle G\times ]0,t_{0}]}$

Numerical methods for parabolic initial-boundary value problems

If the domain of definition of the equation does not change with time, the parabolic initial boundary value problem represents an initial value problem in the time direction and a boundary value problem in the spatial direction for an elliptic differential equation . In the numerical treatment, these two problems can i. w. approach separately. There are two approaches: ${\displaystyle G}$ ${\displaystyle t}$

• Line method ( Engl. MOL = method-of-lines) is discretized first in place by standard methods for elliptic boundary value problems using, as the difference method or finite element methods . This results in a common initial value problem of very large dimensions for the degrees of freedom of the discretization. However, this is a rigid initial value problem and should be solved with implicit or linear-implicit methods, such as the Rosenbrock-Wanner method or the BDF method . The advantage of this approach is that the standard methods mentioned can be used for time integration. The disadvantage is that the location discretization is fixed and therefore local, time-dependent refinements are not possible.
• Rothe method : First, one discretizes in time with one of the methods just mentioned for stiff initial value problems. As a result, an elliptical boundary value problem for the current solution in the area is given in each time step . To solve this boundary value problem z. B. Finite element method with adaptive grid adaptation can be used. However, programming is much more complex than with the line method. ${\displaystyle u(t_{k},{\vec {x}})}$ ${\displaystyle G}$

A simple numerical method for parabolic problems is the Crank-Nicolson method . On the one hand, this uses the finite difference method with a fixed grid for spatial discretization and the implicit trapezoidal method as time discretization .

literature

• Gerhard Dziuk: Theory and Numerics of Partial Differential Equations. de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 , pages 183-253.

Individual evidence

1. ↑ Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 , p. 350.
2. ↑ Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 , pp. 370f.
3. ↑ Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 , pp. 376f.