Partial Differential Equation Solver
• numol(x_endpts, xpts, t_endpts, tpts, num_pde, num_pae, pde_func, pinit, bc_func)
Returns an [xpts x tpts] matrix containing the solutions to the one-dimensional Partial Differential Equation (PDE) in pde_func. Each column represents a solution over one-dimensional space at a single solution time. For a system of equations, the solution for each function is appended horizontally, so the matrix always has xpts rows and tpts * (num_pde + num_pae) columns. The solution is found using the numerical method of lines.
Arguments
• x_endpts, t_endpts are two-element column vectors that specify the real endpoints of the integration regions.
• xpts, tpts are the integer number of points in the integration regions at which to approximate the solution.
• num_pde, num_pae are the integer number of partial differential and partial algebraic equations (PAEs) respectively. num_pde must be at least 1, num_pae can be 0 or greater.
• pde_func is a vector function of x, t, u, ux and uxx of length (num_pde + num_pae). It contains the right-hand sides of the PDEs/PAEs, and assumes that the left-hand sides are all ut. The solution, u, is assumed to be a vector of functions.
If you are working with a system of PDEs, each u on each row of pde_func is defined by a subscript using the index operator, and also the literal subscript operator. For example, u[0 refers to the first function in the system, and ux[1 refers to the first derivative of the second function in the system.
• pinit is a vector function of x of length (num_pde + num_pae) containing initial conditions for each function in the system.
• bc_func is a num_pde * 3 matrix containing rows of the form:
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For Dirichlet boundary conditions
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[bc_left(t)
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bc_right(t)
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"D"]
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or
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For Neumann boundary conditions
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[bc_left(t)
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bc_right(t)
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"N"]
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◦ In the case that the PDE for the corresponding row contains second partial derivatives, both left and right conditions are required.
◦ If only first partial derivatives are present in a particular PDE, then one or the other boundary condition function should be replaced with "NA" and the last entry in the row is always "D."
◦ If no partial derivatives are present for a particular equation in a system, then that row in the matrix is ignored, and can be filled in as ("NA" "NA" "D").
Additional Information
• Algebraic constraints are allowed, for example, 0 = u2(x) + v2(x) − w(x) for all x.
• The number of boundary functions required matches the order of spatial derivative for each PDE, guaranteeing unique solutions.
• Only hyperbolic and parabolic PDEs can be solved using
numol. For an elliptic equation, such as Poisson's equation, use
relax or
multigrid.