dev.nm.analysis.differentialequation.pde.finitedifference.parabolic.dim1.heatequation

Class HeatEquation1D

java.lang.Object

dev.nm.analysis.differentialequation.pde.finitedifference.parabolic.dim1.heatequation.HeatEquation1D

All Implemented Interfaces:

PDE

public class HeatEquation1D
extends Object
implements PDE

A one-dimensional heat equation (or diffusion equation) is a parabolic PDE that takes the following form. \[ \frac{\partial u}{\partial t} = \beta \frac{\partial^2 u}{\partial x^2}, \] The initial condition has this form: \[ u(0,x) = f(x), 0 < x < a \] The mixed boundary conditions have this form: \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t), \\ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \leq c_k \leq 1, k = 1, 2\).

Choose boundary data before discretizing

Use one consistent system of space and time units: beta has units of space squared divided by time, a is the interval length, and T is the terminal time. The constructor requires positive beta, a and T, and c1/c2 in [0,1]. Initial data f is evaluated only at interior spatial points.

For Dirichlet boundaries choose c1 = c2 = 0 and supply endpoint values as functions of time. For pure Neumann boundaries choose c1 = c2 = 1. The left condition uses the negative spatial derivative, while the right uses the positive derivative; the signs must be accounted for when translating flux data. Intermediate c values represent the mixed conditions shown above.

Ensure the initial profile and boundary functions are compatible at time zero when a smooth classical solution is intended. Constructing the PDE stores the problem data; it does not solve or sample the functions.

See Also:

Wikipedia: Parabolic partial differential equation, Wikipedia: Heat equation

Constructor Summary

Constructors

Constructor and Description
HeatEquation1D(double beta, double a, double T, UnivariateRealFunction f, double c1, UnivariateRealFunction g1, double c2, UnivariateRealFunction g2) Constructs a heat equation problem.

Method Summary

Modifier and Type	Method and Description
double	a() Gets the size of the one-dimensional space, that is, the range of x, (0 < x < a).
double	beta() Gets β in the equation (also called thermal diffusivity in case of the heat equation).
double	c1() Gets the coefficient \(c_{1}\) in the mixed boundary condition at the boundary x = 0. \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \] where \(0 \leq c_1 \leq 1\).
double	c2() Gets the coefficient \(c_{2}\) in the mixed boundary condition at the boundary x = a. \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \leq c_2 \leq 1\).
double	f(double x) Gets the initial condition of u at the given position x.
double	g2(double t) Gets the value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = a\) at the given time \(t\). \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \]
double	T() Gets the time period of interest, that is, the range of t, (0 < t < T).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

HeatEquation1D

public HeatEquation1D(double beta,
                      double a,
                      double T,
                      UnivariateRealFunction f,
                      double c1,
                      UnivariateRealFunction g1,
                      double c2,
                      UnivariateRealFunction g2)

Constructs a heat equation problem.

Parameters:

beta - the beta in the equation

a - the region of interest (0, a)

T - the time period of interest (0, T)

f - the initial condition of u, i.e., u(0,x)

c1 - the coefficient in the mixed boundary condition at x = 0

g1 - the mixed boundary condition at x = 0

c2 - the coefficient in the mixed boundary condition at x = a

g2 - the mixed boundary condition at x = a

Method Detail

beta

public double beta()

Gets β in the equation (also called thermal diffusivity in case of the heat equation).

Returns:

β

a

public double a()

Gets the size of the one-dimensional space, that is, the range of x, (0 < x < a).

Returns:

the size of the space

T

public double T()

Gets the time period of interest, that is, the range of t, (0 < t < T).

Returns:

the time period of interest

f

public double f(double x)

Gets the initial condition of u at the given position x.

Parameters:

x - the position

Returns:

u(0, x)

c1

public double c1()

Gets the coefficient \(c_{1}\) in the mixed boundary condition at the boundary x = 0. \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \] where \(0 \leq c_1 \leq 1\). If \(c_1 = 0\), this is the Dirichlet boundary condition. If \(c_1 = 1\), this is the Neumann boundary condition.

Returns:

the coefficient \(c_{1}\)

c2

public double c2()

Gets the coefficient \(c_{2}\) in the mixed boundary condition at the boundary x = a. \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \leq c_2 \leq 1\). If \(c_2 = 0\), this is the Dirichlet boundary condition. If \(c_2 = 1\), this is the Neumann boundary condition.

Returns:

the coefficient \(c_{2}\)

g2

public double g2(double t)

Gets the value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = a\) at the given time \(t\). \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \]

Parameters:

t - the time

Returns:

\(g_2(t)\)