Chapter 6.7, Problem 1E

Apply the Adams-Bashforth Two-Step Method to the IVPs

$\begin{array}{l}\text{a}\text{. }y\text{'}=t\\ \text{b}\text{. }y\text{'}=t^{2}y\\ \text{c}\text{.}y\text{'}=2(t+1)y\\ \text{d}\text{. }y\text{'}=5t^{4}y\\ \text{e}\text{. }y\text{'}=1/y^2\\ \text{f}\text{. }y\text{'}=t^3/y^2\end{array}$

with initial condition $y(0)=1$ . Use step size $h=1/4$ on the interval [0, 1]. Use the Explicit Trapezoid Method to create $w_1$ . Using the correct solution in Exercise 6.1.3, find the global truncation error at $t=1$.

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

  $\begin{array}{ccc}{w}_1& =& w_0+\frac{h}{2}[f(t_0,w_0)+f(t_0+h,w_0+hf(t_0,w_0))]\\ & =& 1+\frac{1}{8}[t_0+t_1]=1+\frac{1}{8}\left[\frac{1}{4}\right]=\frac{33}{32}.\end{array}$

For the remainder of the steps, we apply the Adams-Bash forth Two-Step Method, which is

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h\left[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})\right]\\ & =& w_i+h\left[\frac{3}{2}{t}_i-\frac{1}{2}{t}_{i-1}\right].\end{array}$

Therefore

  $\begin{array}{ccc}{w}_2& =& \frac{33}{32}+\frac{1}{4}\left[\frac{3}{2}\cdot \frac{1}{4}\right]=\frac{9}{8}\\ w_3& =& \frac{9}{8}+\frac{1}{4}\left[\frac{3}{2}\cdot \frac{1}{2}-\frac{1}{2}\cdot \frac{1}{4}\right]=\frac{41}{32}\\ w_4& =& \frac{41}{32}+\frac{1}{4}\left[\frac{3}{2}\cdot \frac{3}{4}-\frac{1}{2}\cdot \frac{1}{2}\right]=\frac{3}{2}\end{array}$

Global error at $t=1$  is 0.

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept Used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

The Adams-Bash forth Two-Step Method is

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})]\\ & =& w_i+h[\frac{3}{2}{t}_i^2{w}_i-\frac{1}{2}{t}_{i-1}^2{w}_{i-1}].\end{array}$

Completing the last three steps results in the following table:

  $\begin{array}{cc}{t}_i& w_i\\ 0& 1.0000\\ 1/4& 1.0078\\ 1/2& 1.0314\\ 3/4& 1.1203\\ 1& 1.3243\end{array}$

Global error at $t=1 \text{is} \mathrm{0.0713.}$

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept Used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})]\\ & =& w_i+h[\frac{3}{2}2(t_i+1)w_i-\frac{1}{2}2(t_{i-1}+1)w_{i-1}].\end{array}$

The Adams-Bash forth approximation

  $\begin{array}{cc}{t}_i& w_i\\ 0& 1.0000\\ 1/4& 1.7188\\ 1/2& 3.0801\\ 3/4& 6.0081\\ 1& 12.7386\end{array}$

Global error at $t=1 \text{is} \mathrm{7.3469.}$

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})]\\ & =& w_i+h[\frac{3}{2}5t_i^4{w}_i-\frac{1}{2}5t_{i-1}^4{w}_{i-1}].\end{array}$

The Adams-Bash forth approximation

  $\begin{array}{cc}{t}_i& w_i\\ 0& 1.0000\\ 1/4& 1.0024\\ 1/2& 1.0098\\ 3/4& 1.1257\\ 1& 1.7540\end{array}$

Global error at $t=1$ is .9642.

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})]\\ & =& w_i+h[\frac{3}{2}\frac{t_i^3}{w_i^2}-\frac{1}{2}\frac{t_{i-1}^3}{w_{i-1}^2}].\end{array}$

The Adams-Bash forth approximation

  $\begin{array}{cc}{t}_i& w_i\\ 0& 1.0000\\ 1/4& 1.0020\\ 1/2& 1.0078\\ 3/4& 1.0520\\ 1& 1.1796\end{array}$

Global error at $t=1 \text{is} \mathrm{0.0255.}$

To determine

To find: the solution for the given IVPs problem for given step.

Explanation of Solution

Concept used:

Dr.Heun refer to an improved or modified Euler method (that is, an explicit trapezoidal rule)." from this it can be understood that the idea of an explicit trapezoidal rule is to replace the correct selection element on the right side with numerical approximation problems solution.

Calculation:

From this theorem known that using the Explicit Trapezoid Method with $y(0)=1andh=1/4$ , we calculate

  $\begin{array}{ccc}{w}_{i+1}& =& w_i+h[\frac{3}{2}f(t_i,w_i)-\frac{1}{2}f(t_{i-1},w_{i-1})]\\ & =& w_i+h[\frac{3}{2}2(t_i+1)w_i-\frac{1}{2}2(t_{i-1}+1)w_{i-1}].\end{array}$

The Adams-Bash forth approximation

  $\begin{array}{cc}{t}_i& w_i\\ 0& 1.0000\\ 1/4& 1.7188\\ 1/2& 3.0801\\ 3/4& 6.0081\\ 1& 12.7386\end{array}$

Global error at $t=1 \text{is} \mathrm{7.3469.}$