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Figure 2: Shooting method for a two-point boundary value problem.3.9.2 Finite difference method[T,Y] = ODE_FINIT_DIFF(RHS, A, B, UA, UB, N) integrates the system of differentialequations u”= f(t,u,u’) from time A to B with boundary conditions u(A) =UA and u(B) = UB on an equispaced grid of N intervals. Function RHS(t,u,u’)must return a column vector corresponding to f(t,u,u’). Each row in thesolution array Y corresponds to a time returned in the column vector T.ParametersRHS integrated function.A T0.B TF.UA initial value Y(T0).UB final value Y(TF).N T equispaced grid of N intervals.ReturnsT equispaced grid of N intervals.Y solution array.3.9.3 Colocation method[T,Y] = ODE_COLLOC(RHS, A, B, UA, UB, DN, N) integrates the system of differentialequations u”= f(t,u,u’) from time A to B with boundary conditions u(A) =UA and u(B) = UB on an equispaced grid of N intervals. Function RHS(t,u,u’)must return a column vector corresponding to f(t,u,u’). Each row in thesolution array Y corresponds to a time returned in the column vector T.25
ParametersRHS integrated function.A T0.B TF.UA initial value Y(T0).UB final value Y(TF).N T equispaced grid of N intervals.DN degree of computed polynomial solution.ReturnsT equispaced grid of N intervals.Y solution array.3.10 Partial Differential EquationsWe turn now to partial differential equations (PDEs), where many of the numerical tech- niqueswe saw for ODEs, both initial and boundary value problems, are also applicable. The situationis more complicated with PDEs, however, because there are additional independent variables,typically one or more space dimensions and possibly a time dimension as well. Additional dimensionssignificantly increase computational complexity. Problem formulation also becomesmore complex than for ODEs, as we can have a pure initial value problem, a pure boundaryvalue problem, or a mixture of the two. Moreover, the equation and boundary data may bedefined over an irregular domain in space.First, we establish some notation. For simplicity, we will deal only with single PDEs (asopposed to systems of several PDEs) with only two independent variables (either two spacevariables, which we denote by x and y, or one space and one time variable, which we denote byx and t). In a more general setting, there could be any number of dimensions and any numberof equations in a coupled system of PDEs. We denote by u the unknown solution function to bedetermined and its partial derivatives with respect to the independent variables by appropriatesubscripts: u x = ∂u/∂x, u xy = ∂ 2 u/∂x∂y, etc.3.10.1 Method of lines (for Heat equation)[T, X, U] = PDE_HEAT_LINES(NX, NT, C, F) solves the heat equation D U/DT= C D2 U/DX2 with the method of lines on [0,1]x[0,1]. Initial conditionis U(0,X) = F. C is a positive constant. NX is the number of space integrationintervals and NT is the number of time-integration intervals.ParametersNX X equispaced grid of NX intervals.NT T equispaced grid of NX intervals.C positive constant.ReturnsT equispaced grid of NT intervals.26
Page 3 and 4: 3.8.2 Implicit Euler . . . . . . .
Page 6 and 7: • Euler• Implicit Euler• Modi
Page 8 and 9: X0 initial guess.ReturnsX computed
Page 10 and 11: 3.2.1 Normal equation[X, RES] = NOR
Page 12 and 13: X0 initial point.ITMAX maximal numb
Page 14 and 15: 3.4.1 Monomial basisP = ITPOL_MONOM
Page 16 and 17: N number of subdivisions.ReturnsRES
Page 18 and 19: 3.6.4 Orthogonal iteration[LAMBDA,
Page 20 and 21: H f (xk)s k = f (x k ) (12)for s k
Page 22 and 23: 3.8.1 Euler[TT,Y] = ODE_EULER(ODEFU
Page 24 and 25: condition Y0 using the fourth-order
Page 28 and 29: X equispaced grid of NX intervals.U
Page 30 and 31: ReturnsT grid of NT intervals.X gri
Page 32: A, BC, D rectangle (A,B)x(C,D) wher

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