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Unformatted text preview: MA2213 Lecture 11 PDE Topics ntroduction p. 451452 oisson equation p. 453466 Boundary conditions p. 453 Finite difference grid p. 454456 MATLAB Program p. 456458 Visualization of numerical results p. 459466 nedimensional heat equation p. 466481 Introduction Many phenomena in sciences and engineering epend on more than one variable. For example, n unknown function of a realworld problem sually depends on both time t and the location f the point (x,y,z).” p. 451 hysical laws, including the conservation of nergy, momentum and mass, the laws of lectricity and magnetism, thermodynamics, nd chemical kinetics, require that the partial erivatives of these functions satisfy certain partial differential) equations. Introduction ncreasingly, PDE’s are used to model biological nd social phenomena. The models include the law of supply and demand” in economics that etermines equilibrium prices of goods and ervices, the BlackSholes equation for options rices in arbitragefree financial markets, and aws that describe the evolution of population ensities that are used in epidemiology, ecology, nd population genetics. http://www.imbs.uci.edu/index.html Introduction xamples 2 2 2 2 2 ) , ( ), , ( f R y x y x y u x u ⊂ Ω ∈ = ∂ ∂ + ∂ ∂ Poisson equation ), , ( ), , ( f 2 2 > ∈ + ∂ ∂ = ∂ ∂ t L x t x x u a t u Heat equation ), , ( ), , ( f 2 2 2 2 > ∈ + ∂ ∂ = ∂ ∂ t L x t x x u a t u Wave equation Poisson Equation Boundary Conditions et be a planar domain, and denote its The boundary value problem 2 R ⊂ Ω oundary by . Ω ∂ Ω ∈ = ∂ ∂ + ∂ ∂ ) , ( ), , ( f 2 2 2 2 y x y x y u x u Ω ∂ ∈ = ) , ( ), , ( g y x y x u s called a Dirichlet problem because the value of...
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 Fall '07
 Michael
 matlab, Numerical Analysis

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