Teaching CSIR NET Mock Test Series Mathematical Science Linear Integral Equations Fredholm and Volterra Integral Equation
The integral equation \(\int_a^x\)k(x, y) ϕ (y) dy = f(x) with K(x, x) ≠ 0; f or all x can be transformed to ϕ(x) + \(\int_a^x\) G(x, y) ϕ (y)dy = g(x), where for all x, y
1
k(x, x) = 1, g = f' and G(x, y) = \(\rm \frac{\partial k}{\partial x}(x, y)\)
2
k(x, y) = 1, g = f and G(x, y) = 0
3
G(x, y) = \(\rm \frac{1}{K(x, x)} \frac{\partial k}{\partial x}(x, y) \) and \(\rm g(x)=\frac{f^{\prime}(x)}{k(x, x)}\)
4
G(x, y) = 1 and g(x) = f(x)