Q9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

Step-by-Step Solution

Answer

The result is: 

$$\begin{gathered} {\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[{\mathbf{f}}_{\mathbf{i}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}\mathbf{+}{\mathbf{f}}_{\mathbf{i}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{+} \\ {\mathbf{hf}}_{\mathbf{1}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{+}{\mathbf{hf}}_{\mathbf{m}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}\right] \end{gathered}$$

Step 1: Use Euler’s method

Here given Euler’s method of the differential equation:

 So, ${\mathbf{y}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{y}}_{\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[\mathbf{f}\mathbf{(}{\mathbf{x}}_{\mathbf{n}}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{)}\mathbf{+}\mathbf{f}\mathbf{(}{\mathbf{x}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{+}\mathbf{hf}\mathbf{(}{\mathbf{x}}_{\mathbf{n}}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{\left)}\mathbf{\right)}\right]$

n=0,1,2,…..and ${\mathbf{x}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{n}}\mathbf{+}\mathbf{h}$.

 Now, 

$$\begin{gathered} \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{1}}\mathbf{(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{)} \\ \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{2}}\mathbf{(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{)} \\ \mathbf{.} \\ \mathbf{.} \\ \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{m}}\mathbf{(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{)} \end{gathered}$$ 

Step 2: Solve for every i

For every I from 1 to m, then;

$$\begin{gathered} {\mathbf{t}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h} \\ {\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[{\mathbf{f}}_{\mathbf{i}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}\mathbf{+}{\mathbf{f}}_{\mathbf{i}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{+} \\ {\mathbf{hf}}_{\mathbf{1}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{+}{\mathbf{hf}}_{\mathbf{m}}\mathbf{(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{)}\right] \end{gathered}$$ 

 

This is the required result.