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Partial Differential Equation

based Models

Topic: INTRODUCTION+ Heat Equation

Mathematical Modeling

& Simulation

INTRODUCTION

General Features of Partial Differential Equations Classification of Partial Differential Equations Classification of Physical Problems Elliptic Partial Differential Equations Parabolic Partial Differential Equations Hyperbolic Partial Differential Equations The Convection-Diffusion Equation Initial Values and Boundary Conditions Well-Posed Problems Summary

 To present the general features of partial

differential equations

 To discuss the relationship between the type of

physical problem being solved,

 To make the classification of the corresponding

governing partial differential equation,

 To figure out type of numerical method required

 To present examples to illustrate these concepts.

Objectives

The dependent variable depends on the physical problem being modeled.

A partial differential equation (PDE) is an equation stating a relationship between a function of two or more independent variables and the partial derivatives of this function with respect to these independent variables.

The dependent variable f is used as a generic dependent variable throughout this course.

In most problems in engineering and science, the independent variables are either space (x, y, z) or space and time (x, y, z, t).

GENERAL FEATURES OF PARTIAL DIFFERENTIAL EQUATIONS

The solution of a partial differential equation is that particular function, f(x, y) or f(x, t), which satisfies the PDE in the domain of interest, D(x, y) or D(x, t), respectively, and satisfies the initial and/or boundary conditions specified on the boundaries of the domain of interest.

In a very few special cases, the solution of a PDE can be expressed in closed form.

In the majority of problems in engineering and science, the solution must be obtained by numerical methods.

GENERAL FEATURES

GENERAL FEATURES

For three independent variables x, y and z, The Laplace Equation is

2 0

2 2

2 2

2 2  

  

  

   z

f y

f x

f f

For four independent variables x, y, z and t, the diffusion Equation is



 

 

  

  

 

 

2

2 2

2 2

2

2

z

f y

f x

f f

f f or

t

t 

 The  is the diffusion coefficient.

For four independent variables x, y, z and t, the Wave Equation is



 

 

  

  

 

 

2

2 2

2 2

2 2

2 2

z

f y

f x

f c f

f c f or

tt

tt

Equations are all second-order partial differential equations.

GENERAL FEATURES OF PARTIAL DIFFERENTIAL EQUATIONS

If the coefficients depend on the dependent variable, or the derivatives appear in a nonlinear form, then the PDE is nonlinear. For example,

0

0 (^2)  

  x y

t x af b f

f f bxf

These equations are homogeneous. The example of non-homogeneous equation is

2 ( , , )

2 2

2 2

2 2 F x y Z z

f y

f x

f f  

  

  

  

This is a non-homogeneous Laplace Equation, which is known as the Poisson Equation.

The non-homogeneous term, F(x, y, z), is a forcing function, a source term, or a dissipation function, depending on the application. The appearance of a non-homogeneous term in a partial differential equation does not change the general features of the PDE, nor does it usually change or complicate the numerical method of solution.

GENERAL FEATURES OF PARTIAL DIFFERENTIAL EQUATIONS

Many physical problems are governed by a system of PDEs involving several dependent variables. For example, the two PDEs

0

0  

  t x

t x Ag B f

af bg

comprise a system of two coupled partial differential equations in two independent variables (x and t) for determining the two dependent variables f(x, t) and g(x, t).

Systems containing several PDEs occur frequently, and systems containing higher-order PDEs occur occasionally.

Systems of PDEs are generally more difficult to solve numerically than a single PDE.

GENERAL FEATURES

The third special case is the system of two general quasi- linear first-order non-homogeneous PDEs in two independent variables, which can be written as

where the coefficients a to d may depend on x, t, f and g; the coefficients A to D may depend on x, t, f, and g.

The non-homogeneous terms e and E may also depend on x, t, f and g.

a ft bfx cgt dgx e

A ft  Bfx Cgt  Dgx  E

Solving a heat equation in Matlab

• Periodic Heat Diffusion in Subsurface Rocks
• Consider the problem of determining the subsurface temperature fluctuations of rock as the result of daily or seasonal temperature variations. As the surface is heated or cooled, the heat will diffuse through the soil and rock. Mathematically, diffusion may be represented by the equation

t

T t z

z

T t z

k

2

2

where T(t, z) is the temperature at time t and depth z, and k is a constant which measures the "diffusivity" of the rock, i.e. how quickly heat moves through the rock. A typical value for k is 10 -6^ m^2 /s.

• Boundary Conditions
• The heat equation is a typical example of what is known as a "partial differential equation."
• A differential equation is any equation in which a function Temperature is not represented directly, but via it's derivative. Partial indicates that there are at least two variables (time and space) in the derivatives.
• In order to solve a PDE numerically, we need to specify boundary conditions.
• The four boundaries of our "space" are the surface (0m), the bottom depth (100m), an arbitrary beginning of time (t = 0), and one year later (t= 1yr).

Top Boundary:

• The top boundary is the simplest, we will simply specify a temperature that varies seasonally, i.e.
• This will create an average annual temperature of 15, an annual low of 5 and an annual high of 25
• Here time is measured in months, and we are not starting on Jan 1.

T(t , 0 ) 15 - 10 sin( 2 t/ 12 )

Left and right boundaries

• The left and right boundary conditions are a bit more problematic, as they require that we already know the temperature at every depth.
• But we do know that the temperature now and exactly one year from now should be the same.

Finite difference approximations

• The final step is to determine how to use finite differences to evaluate the derivatives. For the first derivative of temperature with respect to time we will use the "forward derivative"

t

T t z T t z t

T ti z i i 

 

 ( , ) (  1 , ) ( , )

For the first derivative of temperature with respect to depth, we will use the "central derivative"

z

T t z T t z z

T t zj j i j 

 

 (^)  - 2

( , ) ( , 1 ) ( , 1 )