Picard’s iterative method helps to solve the differential equation by a sequence of approximations as to the solution in which the nth approximation depends on the previous approximations.
It is an iterative method used for approximating the solutions of differential equations. The solution of the differential equation will be more accurate if it is used repeatedly. It is easier to implement this method and the solution generally in power series.
If we consider the first-order differential equation as:
Integrating between limits, we get
Now, by replacing as
The second approximation is:
Continuing the process, the nth approximation is:
This is known as recursion or iterative formula.
Now, this Picard method generates a sequence of approximations as which converges to the exact solution y(x). So the function f(x,y) is bounded in the neighborhood of the point and it satisfies the Lipschitz conditions.
A function f(x)
The solution after performing picard’s method
Step 1: [defining y1] return (1 + (x) + x^2/2) Step 2: [defining y2] return (1 + (x) + x^2/2 + x^3/3 + x^4/8) Step 3: [defining y3] return (1 + (x) + x^2/2 + x^3/3 + x^4/8 + x^5/15 + x^6/48) step 4:[Picard’s method] read x0, xn [the lower and upper limit of the integration] and the allowed error Set c ← 0 for tmp = x0 to xn repeat Set y1[c] ← f1(tmp) Set y2[c] ← f2(tmp) Set y3[c] ← f3(tmp) Set tmp ← tmp + err Set c ← c + 1 [End of ‘for’ loop] [printing the value of x] for tmp = x0 to xn repeat Print tmp Set tmp ← tmp + err [End of ‘for’] [printing the values of y1] Set c ← 0 for tmp = x0 to xn repeat print y1(c) Set tmp ← tmp + err Set c ← c + 1 [End of ‘for’ loop] [printing the values of y2] Set c ← 0 for tmp = x0 to xn repeat print y2(c) Set tmp ← tmp + err Set c ← c + 1 [End of ‘for’ loop] [printing the values of y3] Set c ← 0 for tmp = x0 to xn repeat print y3(c) Set tmp ← tmp + err Set c ← c + 1 [End of ‘for’ loop]
1. It is a straightforward approach.
2. It converges to a unique solution for a certain region.
1. The computation of this method is heavy. It requires a lengthy calculation.
1. It is used to solve the differential equation with an accurate solution.