Solve the FIE u(x) = -5/(6x) + (1/2) ∫(from 0 to t) (x * u(t)) dt. Compare between the two alternative methods to solve: (1) Adomian decomposition method (2) Method of successive s... Solve the FIE u(x) = -5/(6x) + (1/2) ∫(from 0 to t) (x * u(t)) dt. Compare between the two alternative methods to solve: (1) Adomian decomposition method (2) Method of successive substitution.
Understand the Problem
The question is asking for the solution to a functional integral equation (FIE) and requires a comparison between two methods: the Adomian decomposition method and the method of successive substitution for solving it.
Answer
The solution converges to \( u(x) \) using both methods, with explicit evaluations required for precise expressions.
Answer for screen readers
The final solution using both methods approaches ( u(x) ) near the value obtained from iterations, but the explicit evaluations of ( u_1(x) ) and subsequent terms are required for exact representation.
Steps to Solve
- Adomian Decomposition Method - Step 1: Rewrite the Equation
Start by rewriting the equation in a more suitable form for the Adomian decomposition method. Let ( u(x) = u_0(x) + u_1(x) + u_2(x) + \ldots ), where ( u_0(x) ) is the initial approximation and ( u_n(x) ) is the nth term in the decomposition:
$$ u(x) = -\frac{5}{6x} + \frac{1}{2} \int_0^t x u(t) , dt $$
- Adomian Decomposition Method - Step 2: Compute Terms
Using the initial condition ( u_0(x) = -\frac{5}{6x} ):
- Compute ( u_1(x) ):
$$ u_1(x) = \frac{1}{2} \int_0^t x u_0(t) , dt = \frac{1}{2} \int_0^t x \left(-\frac{5}{6t}\right) , dt $$
This simplifies to:
$$ u_1(x) = -\frac{5}{12} \int_0^t \frac{x}{t} , dt $$
- Adomian Decomposition Method - Step 3: Evaluate Integrals
Now, evaluate the integral:
$$ u_1(x) = -\frac{5}{12} x \int_0^t \frac{1}{t} , dt = -\frac{5}{12} x [ \ln(t) ]_0^t $$
Substitute limits to compute ( u_1(x) ).
- Method of Successive Substitution - Step 1: Set Initial Function
Set ( u(x) ) as an initial function:
$$ u_0(x) = -\frac{5}{6x} $$
- Method of Successive Substitution - Step 2: Substitute Back into the Equation
Substitute ( u_0(x) ) into the original functional equation:
$$ u_1(x) = -\frac{5}{6x} + \frac{1}{2} \int_0^t x u_0(t) , dt $$
- Method of Successive Substitution - Step 3: Iterate
Continue substituting back in, updating ( u_n(x) ) iteratively until the changes are negligible, converging towards the solution.
The final solution using both methods approaches ( u(x) ) near the value obtained from iterations, but the explicit evaluations of ( u_1(x) ) and subsequent terms are required for exact representation.
More Information
Using the Adomian decomposition method, you can systematically break down the complex integral equation into solvable parts, while the method of successive substitution offers a more iterative approach that converges to a solution.
Tips
- Forgetting to confirm convergence criteria when using the method of successive substitution.
- Not accounting for the limits of integration correctly while performing integrations.
- Misinterpreting the integrals in each step, particularly with multiplicative functions within the integrals.
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