Convert the initial value problem to a FIE: y''(x) + y(x) = cos x, y(0) = 0, y'(0) = 1. Solve the FIE u(x) = 5/σ x + 1/2 ∫ from 0 to t u(t) dt. Compare between the two alternative... Convert the initial value problem to a FIE: y''(x) + y(x) = cos x, y(0) = 0, y'(0) = 1. Solve the FIE u(x) = 5/σ x + 1/2 ∫ from 0 to t u(t) dt. Compare between the two alternative methods to solve: 1) Adomian decomposition method 2) Method of successive substitution. Convert the FIE into BVP u'' + 7u = 0, u(0) = 0, u(1) = 0. Hint: use Leibniz's rule. Read more

Steps to Solve

1. Understand the Initial Value Problem (IVP)

The initial value problem given is:

$$ y''(x) + y(x) = \cos(x), \quad y(0) = 0, \quad y'(0) = 0. $$

This is a second-order ordinary differential equation.

2. Convert IVP to Fredholm Integral Equation (FIE)

The conversion from the IVP to the FIE format involves expressing the solution in terms of an integral involving a kernel $k(x,t)$:

$$ u(x) = \frac{5}{6}x + \frac{1}{2} \int_0^t x t u(t) , dt. $$

3. Apply the Adomian Decomposition Method

Using the Adomian decomposition method, the solution $u(x)$ can be expressed as a series:

$$ u(x) = \sum_{n=0}^{\infty} u_n(x). $$

Each term in the series represents contributions from the non-linear integral equation.

4. Implement the Method of Successive Substitution

For this method, substitute an initial guess into the integral equation progressively:

$$ u^{(n+1)}(x) = \frac{5}{6}x + \frac{1}{2} \int_0^x k(x,t) u^{(n)}(t) dt, $$

where $u^{(0)}(x) = 0$ is used as the initial guess.

5. Convert the FIE into a Boundary Value Problem (BVP)

The FIE can be expressed in BVP form. For the provided function $k(x,t)$:

$$ u'' + 2u = 0, \quad u(0) = 0, \quad u(1) = 0. $$

The boundary value formulation will reveal solutions fitting the boundary constraints.