## Picard Method

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### Description

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.

DERIVATION

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.

GRAPHICAL REPRESENTATION

### Algorithm

INPUT:

A function f(x)

OUTPUT:

The solution after performing picard’s method

PROCESS:

``````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]``````

### Code

1. It is a straightforward approach.

2. It converges to a unique solution for a certain region.