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L11_PDE - MA2213 Lecture 11 PDE Topics Introduction p...

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MA2213 Lecture 11 PDE
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Topics Introduction p. 451-452 Poisson equation p. 453-466 Visualization of numerical results p. 459-466 Boundary conditions p. 453 Finite difference grid p. 454-456 MATLAB Program p. 456-458 One-dimensional heat equation p. 466-481
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Introduction “Many phenomena in sciences and engineering depend on more than one variable. For example, an unknown function of a real-world problem usually depends on both time t and the location of the point (x,y,z).” p. 451 Physical laws, including the conservation of energy, momentum and mass, the laws of electricity and magnetism, thermodynamics, and chemical kinetics, require that the partial derivatives of these functions satisfy certain (partial differential) equations.
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Introduction Increasingly, PDE’s are used to model biological and social phenomena. The models include the “law of supply and demand” in economics that determines equilibrium prices of goods and services, the Black-Sholes equation for options prices in arbitrage-free financial markets, and laws that describe the evolution of population densities that are used in epidemiology, ecology, and population genetics. http://www.imbs.uci.edu/index.html
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Introduction Examples Wave equation 2 2 2 2 2 ) , ( ), , ( f R y x y x y u x u = + Poisson equation 0 ), , 0 ( ), , ( f 2 2 + = t L x t x x u a t u Heat equation 0 ), , 0 ( ), , ( f 2 2 2 2 + = t L x t x x u a t u
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Poisson Equation Boundary Conditions Let be a planar domain, and denote its The boundary value problem 2 R boundary by . = + ) , ( ), , ( f 2 2 2 2 y x y x y u x u = ) , ( ), , ( g y x y x u is called a Dirichlet problem because the value of u is specified on the boundary. For simplicity we
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