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Partial Differential Equations

LABWORK 3

COMPUTATIONAL VERSUS ANALYTICAL SOLUTIONS OF

1-DIMENSIONAL HOMOGENEOUS

HEAT BOUNDARY VALUE PROBLEMS

1. Key in the following Maple commands then answer the following questions.

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a) What is the BVP that is being soled in the program segment aboe.

Ans:!1! >

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Partial Differential Equations

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d) *+etch the solution cures obtained computationally by the command

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e) ,nterpret this isual.

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&his figure shows the heat profile along the rod for t>0.,nitially the heat at eerypoint along the rod is goerned by the equation ( ) ( )sin $ # sin "x x . -fter!.!!" sec the temperature profile still appears sinusoidal but with the amplitude

reduced. -fter !.!1 sec the temperature further reduces getting closer to !. -fter

!.!" sec the temperature along the rod is een closer to !. -fter about !.1 sec the

temperature along the rod is still sinusoidal. -fter 1 sec the heat stabilies to !

temperature. /oweer at all times the ends of the rod are unchanged and remain !.

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k=1.0

x=0 x=1

not insulated

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Partial Differential Equations

f) *+etch and interpret the animation from the command

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sing *2V the BVP with %irichlet boundary conditions has analytical solution as

follows.

)"sin3#)$sin3)sin3)(3### )"3)$3

1

)3xexexneAtxu

tt

n

tn

n

=

==

g) &ype out the following Maple commands.

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1"

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Partial Differential Equations

h) *+etch and compare 4igure $ with 4igure 1.

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i) *+etch and compare 4igure # and 4igure 0.

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Partial Differential Equations

#. 6onsider the following heat boundary alue problem.

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b) %escribe in words what the model represent.

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c) *+etch the physical representation of the BVP.

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Partial Differential Equations

d) *+etch the solution cures obtained computationally by the Maple command.

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e) ,nterpret this isual.

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f) ,nterpret the animation obtained from

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Partial Differential Equations

&his BVP with %irichlet boundary conditions has analytical solution as follows.

( )nxeAtxu tnn

n

sin)(3#

1

=

=

where ( ) ( )

== ndxnxA

n

n11#!!sin#!!

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g) &ype out the following Maple commands.

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h) *+etch and compare 4igure $ and 4igure 1.

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Partial Differential Equations

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i) *+etch and compare 4igure # and 4igure 0.

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:) ;ow add the following Maple commands then answer the following

questions.

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#!

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Partial Differential Equations

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+) *+etch the isual obtained from the segment aboe.

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l) ,nterpret this isual.

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#1

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Partial Differential Equations

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