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Heat_Equation

The 2-d heat equation is solved using partial differential equations generated using the Modular Program Constructor (MPC). Heat_Equation_ODE solves the problem using ordinary differential equations.

Model number: 0363

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Description

 Heat equation in two dimensions is solved using partial differential
equations.



Contour plots of the solution of heat equation using the prescribed Dirichlet boundary conditions at Time = 0.01, 0.02, 0.04 and 1.00 seconds.

Equations

Partial Differential Equation

$\large {\frac {\partial C}{\partial t}}= D \cdot \left( {\frac {\partial ^{2}C}{\partial {x}^{2}}}+ {\frac {\partial ^{2} C}{\partial {y}^{2}}} \right)$

Boundary Conditions

$\large {C \left( x,0 \right) =\sin \left( \pi \,x \right) }$
$\large {C \left( 0,y \right) =\sin \left( \pi \,y \right) }$
$\large {C \left( x,1 \right) = 0}$
$\large {C \left( 1,y \right) = 0 }$

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

References




Key Terms

Heat equation, 2-D, partial differential equation, PDE, ordinary differential equation, ODE, Modular Program Constructor, MPC

Model History

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Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.