Lec 302 W7 Heat Wave

8/20/2019 Lec 302 W7 Heat Wave

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Partial Differential EquationsHeat Equation

Wave Equation

Laplace Equation

Dr. Ahmed Sayed AbdelSamea

Giza, Egypt, Fall 2015

[email protected]


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Introduction

Linearity

• A linear operator satisfies (for any two functions and ):( + ) () + ()

where and are arbitrary constants.• Any linear combination of linear operators is a linear operator.

• A linear equation for is given in the form , where isknown.

Example: , and are linear operators.Notes: For Linear PDEs, the dependent variable and its partial

derivatives appear only to the first power.


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Introduction

Homogeneity

For , if , it becomes and it is called linearhomogenous equation.

Example:

is a linear homogenous PDE.

Superposition Principle

If , , … , are solutions to a linear homogenous PDE, then

+ + ⋯ + =

where , , , … , are arbitrary constants, is also a solution.


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The Heat Equation

Derivation of the Heat Equation includes:

• Thermal energy density.

• Conservation of heat energy.

• Heat flux and heat sources.

• Fourier’s law of heat conduction.

• Fick’s law of diffusion.

Consider the model of heat flow through a thin insulated wirewhose ends are kept at fixed temperature and its initial

temperature distribution is given.


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The Heat Equation

The Heat Equation (Initial-Boundary Value Problem):

+

(,) , < < , >

ICs:

, , < <

BCs: , , ,


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The Heat Equation

•

, is the temperature at a point

and a time

of a thin wire

of length ,• is the thermal diffusivity (/).•

is the thermal conductivity.

• is the specific heat capacity.• is the mass density of the wire material.•

(,) is the internal heat source within material.• is the initial temperature distribution along the wire.• and are the temperatures at and respectively.


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The Heat Equation

Initial Conditions

, , < <

where is the temperature at each point along the wire at theinitial time.

Boundary Conditions

• Dirichlet Conditions: (Fixed BCs) , , , .• Neumann Conditions: (Insulated BCs) , , .•

Robin Conditions: (Poorly Insulated BCs) , , , > , , , >

where is the outside temperature.


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Heat Equation with Fixed BCs

Consider the heat equation without source of heat (linear

homogenous PDE with linear homogenous BCs):

, < < , >

ICs: , , < < BCs: , , >

(Dirichlet BC)

Using Separation of Variables method, let , ():

′

()() ′′

() () , ICs: , (), < <

BCs: , >


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→ ′

+

→ −

′′ + , Sturm-Liouville problem)→: , ( > )

:

, , , , … Then −(

)

→ , −(

)

, −( )

∞

=


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Separation of Variables, Eigenvalues and Eigenfunctions


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Example Find the solution to the heat flow problem:

, < < , > , , ; > ,

, , < < .

Solution By comparison, we get , then

, −( )

∞

=

→ , ∞

=

→ ,


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, −( )

∞

=

→ , ∞=

→ ,

, −()

+ −()

→ , − − .


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, < < , > , , > ,

, , < < .

Solution

, −( )

∞

=

The Fourier sine series for , → ,

∞

= cos


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Solution

cos /

→ , sin + sin + sin + ⋯ , −(

)

∞

=

, − \ sin +

− \

sin + ⋯


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/


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, < < , > , , > ,

, ≤ ≤ /

/ ≤ ≤

Solution By comparison, we get ,The Fourier sine series for ,

, ∞

= , sin( ) → , sin

sin+

sin⋯


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, −( )

∞

=

→ , − ∞=

,

−

sin

−

sin+

−

sin⋯


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Heat Equation with Insulated BCs

Consider the heat equation without source of heat (linear

homogenous PDE with linear homogenous BCs):

, < < , >

ICs: , , < < BCs: , , > (Neumann BC) Using Separation of Variables method, let , () :

′

()() ′′

() () , ICs: , (), < <

BCs: , >


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Heat Equation with Insulated BCs

→ ′

+

→ −

′′ + , Sturm-Liouville Prob.)→: , ( ≥ )

: , , , , , … Then −(

)

→ , −( )

, + −( )

∞

=


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Heat Equation for thin circular wire

Consider the heat equation for thin circular wire with lateral

insulated side (assume the length is and is the arc length):

, < < , >

ICs: , , < < BCs: , , >

, , The solution can be given by:

, + −( )

∞

=+ −(

)


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The Wave Equation

The Wave Equation (Initial-Boundary Value Problem):

, < < , > ICs: , , , () ≤ ≤ BCs: , , , >


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The Wave Equation

•

, the displacement (deflection) of a string or a wire that is

stretched between two fixed points (Violin, guitar, cello,…) .

• is a positive constant and depends on the linear density andthe tension of the string

(

/).

• The BCs: , , , reflect the fact that the stringis fixed at and .

• The ICs represents the initial displacement , andthe initial velocity , () of the string.

Note that and .


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Separation of Variables for the Wave Equation

Using Separation of Variables method, let

, ():

′′()()

′′() () ,

→ ′′

+

′′

+

′′ + , Sturm-Liouville problem)

, , , , …

′′ + +


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→ , + The formal solution of the Wave Equation is given by:

,

+

∞

=

where

()sin

, : ,

()sin

, : , .


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Notes

• The solutions are periodic in time with frequency arecalled standing waves or normal modes.

• All of these frequencies are integer multiple of the fundamental frequency and for > are called harmonics. Thelarger the frequency, the higher the pitch of the sound produced.

• The blend of fundamental frequency with higher harmonics

gives the pleasing sound of the vibrating string.


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Standing waves

• The normal mode (harmonic) composed of the product:

+

+ where + and tan−(/) . These arestanding wave travels with time-varying amplitude.

• The points in (,) for which / , correspond topoints on a standing wave where there is no motion are called

nodes.


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Standing waves


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Traveling Waves

Every standing wave can be decomposed into two traveling waves.

For example, if we consider the term:

cos

()

cos

(+)

This is called D’Alembert’s solution and can be reached by

substituting

+ and

into the wave equation

( , ) (,).The D’Alembert’s solution can be written as

, + + ( )


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Example

Suppose that the string is stretched and fixed at and .the string is plucked in the middle which means that its shape is

described by ≤ ≤ /

/ ≤ ≤ . At the string is

released with initial velocity . Find the displacement ofthe spring , assuming that . /. Solution

By comparison, we get . , then ,

+

∞

=

→


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Solution

()sin

sin /

+ ( ) sin

/ sin( )

, . ∞

=

sin . sin . +

sin . + ⋯

Lec 302 W7 Heat Wave

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