###### Numerical Methods: Problems and Solutions

333 SOLVED PROBLEMS

Question: 5.31

## Consider an implicit two-step method yn + 1 – (1 + a) yn + a yn – 1 = h/12[(5 +a) yn + 1′ + 8(1 − a)yn′ − (1 + 5a) yn− 1′] where – 1 ≤ a < 1, for the solution of the initial value problem y′ = f (x, y), y(x0) = y0. (i) Show that the order of the two-step method is 3 if a ≠ – 1 and is 4 if ...

(i) The truncation error of the two-step method is...
Question: 5.55

## Use the shooting method to solve the mixed boundary value problem u″ = u – 4x e^x, 0 < x < 1, u(0) – u′(0) = – 1, u(1) + u′(1) = – e. Use the Taylor series method uj+1 = uj + huj′ + h²/2 uj″ + h³/6 uj′″. uj+ 1′ = uj′ + huj″ h²/2  + uj′″ to solve the initial value problems. Assume h = 0.25. Compare ...

We assume the solution in the form u(x)=u_{...
Question: 5.58

## Solve the boundary value problem y″ + (1 + x²) y + 1 = 0, y (± 1) = 0 with step lengths h = 0.5, 0.25 and extrapolate. Use a second order method. ...

Replacing $x$ by $-x$, t...
Question: 5.56

## Use the shooting method to find the solution of the boundary value problem y″ = 6y², y(0) = 1, y(0.5) = 4 / 9. Assume the initial approximations y′(0) = α0 = – 1.8, y′(0) = α1 = – 1.9, and find the solution of the initial value problem using the fourth order Runge-Kutta method with h = 0.1. Improve ...

We use the fourth order Runge-Kutta method to solv...
Question: 5.57

## Use the Numerov method with h = 0.2, to determine y(0.6), where y(x) denotes the solution of the initial value problem y″ + xy = 0, y(0) = 1, y′(0) = 0. ...

The Numerov method is given by  \begin{ali...
Question: 5.66

## (a) Determine the constants in the following relations : h^–4δ^4 = D^4(1 + aδ² + bδ^4) + O(h^6), hD = μδ + a1Δ³ E^–1 + (hD)^4(a2 + a3 μδ + a4δ²) + O(h^7). (b) Use the relations in (a) to construct a difference method for the boundary value problem y^iv (x) = p(x)y(x) + q(x) y(0), y(1), y′(0) and ...

(a) Applying the difference operators on y\...
Question: 5.67

## The differential equation y″ + y = 0, with initial conditions y(0) = 0, y(h) = K, is solved by the Numeröv method. (a) For which values of h is the sequence { yn }0^∞ bounded ? (b) Determine an explicit expression for yn . Then, compute y6 when h = π/6 and K = 1/2. ...

The Numeröv method  y_{n+1}-2 y_{n}+y_{n-1}...
Question: 5.68

## A diffusion-transport problem is described by the differential equation for x > 0, py″ + V y′ = 0, p > 0, V > 0, p/V << 1 (and starting conditions at x = 0). We wish to solve the problem numerically by a difference method with stepsize h. (a) Show that the difference equation which arises when ...

(a) Replacing the derivatives y^{\prime \pr...
Question: 5.69

## In order to illustrate the significance of the fact that even the boundary conditions for a differential equation are to be accurately approximated when difference methods are used, we examine the differential equation y″ = y, with boundary conditions y′(0) = 0, y(1) = 1, which has the solution ...

(i) The difference equation at x=x_{n}[/lat...