Numerical Analysis - Math Problem Example

For the following equate first-ordered differential equation) Runge-Katta can be limited to the Euler method:

If m=1:
(1)

error approximation will be assume that the following condition is followed solution of the equation is differentiated on

Then:

Putting (4.3) into (4.2) we get:

CCondition1 is correct if the following conditions are followed for this function:

Reflection of  in the replacive system (1) is invariantly continuous in functions’ domain.

The order of Euler’s approximation can be cleared if the following condition is followed: solution of x(t) system (1) has limited second derivative on . So that there is a constant C>0 for which  if .

In this case series (3) are defined as follows:

Implies that   and it’s the first degree of approximation. Using any computerized method, the solution of the equation: x1 =DDT DDTt =it can be found using the following algorithm can be generated on any programming language. 

The equation given in the task was solved by the Mathcad program using the program module which solving solve differential equations with fixed step: F: = refixed (Z0, t0, tk, N, f). The result of the solution of this equation in Mathcad is the following:

To evaluate these results, we can solve the same equation using conventional means. As it’s shown this equation is solved by the method of variables separation. After finding the function we should plug the values of t into this function and find its values for all values of t on the interval [0,1].

As we can see the results of the numerical solution of the Runge Kutta method are very close to the real results of this function.

Using error evaluation method:

Absolute value (real value of function- approximated value)/ real value of function) we will get the following results:

Real values:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1.005013

1.020201

1.046028

1.083287

1.133148

1.197217

1.277621

1.377128

1.499303

1.648721

Errors:

0

1.25E-05

0.000197

2.66E-05

0.000265

0.000131

0.000182

0.0003

9.28E-05

0.000202

0.00017

As we can see the results are very reliable as the error is less than 0.001% for all values got.