The Picard’s method is an iterative method and is primarily used for approximating solutions to differential equations.
This method of solving a differential equation approximately is one of successive approximation; that is, it is an iterative method in which the numerical results become more and more accurate, the more times it is used.
The Picard’s iterative method gives a sequence of approximations Y1(x), Y2(x), …Yk(x) to the solution of differential equations such that the nth approximation is obtained from one or more previous approximations.
The Picard’s iterative series is relatively easy to implement and the solutions obtained through this numerical analysis are generally power series.
Picard’s iteration method formula:
- Step 1: An approximate value of y (taken, at first, to be a constant) is substituted into the right hand side of the differential equation:
dy/dx= f(x, y).
- Step 2: The equation is then integrated with respect to x giving y in terms of x as a second approximation, into which given numerical values are substituted and the result rounded off to an assigned number of decimal places or significant figures.
- Step 3: The iterative process is continued until two consecutive numerical solutions are the same when rounded off to the required number of decimal places.
We may proceed as follows:
where x0 = 0. Hence:
where y0 = 0. which becomes:
- First Iteration:
We do not know y in terms of x yet, so we replace y by the constant value y0 in the function to be integrated.
The result of the first iteration is thus given, at x = 0.3, by:
- Second Iteration:
Now, we use:
The result of the second iteration is thus given by:
- Third Iteration:
Now we use:
The result of the third iteration is thus given by:
at x = 0.3.
- Hence, y = 0.0451, correct upto four decimal places, at x = 0.3.
Program for Picard’s iterative method:
X 0.0000 0.4000 0.8000 1.2000 1.6000 2.0000 2.4000 2.8000 Y(1) 1.0000 1.4800 2.1200 2.9200 3.8800 5.0000 6.2800 7.7200 Y(2) 1.0000 1.5045 2.3419 3.7552 6.0645 9.6667 15.0352 22.7205 Y(3) 1.0000 1.5053 2.3692 3.9833 7.1131 13.1333 24.3249 44.2335
- Program for Gauss Siedel Method (Computational Mathematics)
- Iterative program to generate distinct Permutations of a String
- Space efficient iterative method to Fibonacci number
- Write an iterative O(Log y) function for pow(x, y)
- Mathematics | Generalized PnC Set 1
- Mathematics | Generalized PnC Set 2
- Mathematics | Probability
- Power in Mathematics
- Mathematics | Rules of Inference
- Mathematics | Law of total probability
- Mathematics | Introduction of Set theory
- Mathematics | Predicates and Quantifiers | Set 1
- Mathematics | Propositional Equivalences
- Mathematics | Power Set and its Properties
- Mathematics | Conditional Probability
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.