INTEGRAL EQUATIONS SM-4325
Find the solution, for any λ ∈ ℝ, of the integral equation:
φ(s) = 1 + λ ∫_{-1}^1 (3s + t)φ(t)dt
Solve (1), for λ ≠ ±(1)/(2), using Fredholm's iteration scheme. Hint:
Γ(s,t;λ) = (∑_{p=0}^∞ (-λ)^p/p!C_p(s,t))/(∑_{p=0}^∞ (-λ)^p/p!c_p),
where c_0 = 1, C_0(s,t) = K(s,t), c_p = ∫_a^b C_{p-1}(s,s)ds and C_p(s,t) = c_pK(s,t) - ∫_a^b K(s,x)C_{p-1}(x,t)dx.
3. Prove that the integral equation:
2φ(s) = 1 + ∫_0^1 (φ^2(s) + φ(t))dt.
has no real solution.
4. Find the integral equation that is equivalent to the ordinary differential equation:
φ''(s) + sφ'(s) + sφ(s) = 2, φ(0) = 0, φ'(0) = 1.
Propose a method for solving the integral equation.
INTEGRAL EQUATIONS SM-4325
1. Find the solution, for any λ ∈ ℝ, of the integral equation
φ(s) = 1 + λ ∫_{-1}^1 (3s + t)φ(t)dt
(1)
2. Solve (1), for λ ≠ ±(1)/(2), using Fredholm's iteration scheme. Hint: Γ(s,t;λ)
where c_0 = 1, C_0(s,t) = K(s,t), c_p = ∫_a^b C_{p-1}(s,s)ds and C_p(s,t) = c_pK(s,t) - ∫_a^b K(s,x)C_{p-1}(x,t)dx.
3. Prove that the integral equation:
2φ(s) = 1 + ∫_0^1 (φ^2(s) + φ(t))dt
has no real solution.
4. Find the integral equation that is equivalent to the ordinary differential equation:
φ''(s) + sφ'(s) + sφ(s) = 2, φ(0) = 0, φ'(0) = 1.