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# Numerical analysis - Math Problem Example

## Extract of sample Numerical analysis

This method of numerical integration finds solutions in the form of resultant solutions.
The equation given in the task was solved by Mathcad program using program module which allows to solve differential equations with fixed step: F := rkfixed(Z0, t0, tk, N, f). The result of the solution of this equation in mathcad is the following:
In order 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 it's values for all values of t on the interval [0,1].

As we can see the results of numerical solution of 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 error is less than 0.001% for all values ...Show more
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## Summary

The solution of differential equations often can not be limited to simple trivial methods. A number of equations can be solved analytically including systems of equations and equations of higher degree. But nevertheless there exist a big number of real problems where differential equations which describe the state of real-time processes can not be solved using simple differentiation and integration rules…
Author : pgottlieb
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