Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODS(b) Substitute them into the predictor equation and, by making that expressionfor ȳ n+1 coincide with the true Taylor series for y n+1 up to order h 3 , establishsimultaneous equations that determine the values of a 1 ,a 2 and a 3 .(c) Find the Taylor series for f n+1 and substitute it and that for f n−1 into thecorrector equation. Make the corrected prediction for y n+1 coincide with thetrue Taylor series by choosing the weights b 1 ,b 2 and b 3 appropriately.(d) The values of the numerical solution of the differential equationdy 2(1 + x)y + x3/2=dx 2x(1 + x)at three values of x are given in the following table:x 0.1 0.2 0.3y(x) 0.030 628 0.084 107 0.150 328Use the above predictor–corrector scheme to find the value of y(0.4) andcompare your answer with the accurate value, 0.225 577.27.18 If dy/dx = f(x, y) then show thatd 2 fdx = ∂2 f2 ∂x +2f ∂2 f2 ∂x∂y + ∂2 ff2∂y + ∂f ( ) 2∂f ∂f2 ∂x ∂y + f .∂yHence verify, by substitution and the subsequent expansion of arguments inTaylor series of their own, that the scheme given in (27.79) coincides with theTaylor expansion (27.68), i.e.y i+1 = y i + hy (1)i+ h2iup to terms in h 3 .27.19 To solve the ordinary differential equation2! y(2)+ h33! y(3) i+ ··· ,du= f(u, t)dtfor f = f(t), the explicit two-step finite difference schemeu n+1 = αu n + βu n−1 + h(µf n + νf n−1 )may be used. Here, in the usual notation, h is the time step, t n = nh, u n = u(t n )and f n = f(u n ,t n ); α, β, µ, andν are constants.(a) A particular scheme has α =1,β =0,µ=3/2 andν = −1/2. By consideringTaylor expansions about t = t n for both u n+j and f n+j , show that this schemegives errors of order h 3 .(b) Find the values of α, β, µ and ν that will give the greatest accuracy.27.20 Set up a finite difference scheme to solve the ordinary differential equationx d2 φdx + dφ2 dx =0in the range 1 ≤ x ≤ 4, subject to the boundary conditions φ(1) = 2 anddφ/dx =2atx =4.UsingN equal increments, ∆x, inx, obtain the generaldifference equation and state how the boundary conditions are incorporatedinto the scheme. Setting ∆x equal to the (crude) value 1, obtain the relevantsimultaneous equations and so obtain rough estimates for φ(2),φ(3) and φ(4).Finally, solve the original equation analytically and compare your numericalestimates with the accurate values.1036