Course Code
mth423


Question
The solution to \(\phi(x) = \lambda\int_{0}^{1}(1 + xt)\phi(t)\mathrm {d}t + f(x)\) is ________

Answer
\(4+ \sqrt{13} - (7 + \sqrt{52})x\)



Question
The set of orthogonal system \(\phi_{n}\) is said to be complete if \(Lim_{n\rightarrow\infty}\int_{I}(f(x) - P)^2\mathrm {d}x = 0\). What is P?

Answer
\(P = (\sum\alpha_{n}\phi_{n})\)



Question
A kernel is said to be positive if T(\phi, \phi) = \(\int\int k(x,y)\phi(x)\phi(y)\mathrm {d}x\mathrm {d}y > 0 \forall \phi\), suh that _____

Answer
\(\int\phi^2(x)\mathrm {d}x \neg 0\)



Question
Suppose k(x,y) = \(\sum_{n=1}^{\infty}\frac {\phi_{n}(x)\phi_{n}(y)}{\lambda_{n}}\), then k(x,y) is ________

Answer
positive definite



Question
Consider the kernel T in the square\(0\leq x\leq 1, 0\leq y \leq 1 where T(x,y)= (1 - x)y, 0\leq y\leq x\leq 1, T(x,y) = (1 - y)x, 0\leq x\leq y\leq 1\). Find the leilinearformular given by T(x, y)

Answer
\(T(x,y) = 2\sum_{n=1}^{\infty}\frac{sin n\pi x sin n\pi y}{n^2\pi^2}\)



Question
If f(t) is throughout piecewise continuous, bounded variation and exponential order, i.e. there exists M_{0}, so, such that \(f(t)\leq M_{0}e^(sot)\), and if we define \(F(s) = \int_{0} ^{\infty}(e^9-st)f(t)\mathrm {d}t\), S may be complex, then F(s) is known as _______of f

Answer
Laplace Transforms



Question
Let \(\phi_{n}\) be an orthogonal system, and let f be continuous, when we set\(\alpha_{n} = \int_{I}f(x)\phi_{n}(x)\mathrm {d}x, then \sum\alpha_{n}\leq\) _________

Answer
\(\int_{I}f^2(x)\mathrm {d}x\)



Question
One of the following is not correct if k(x,y) is symmetric and continous

Answer
The eigenvalues are complex



Question
Let k(x,y) be symmetric and k(x,y) = k(y,x )( and do not identically Zero. Then k has at least one ___

Answer
Eigen value



Question
Find the eigenvalues and eigenfunctions of the system defined by \(\phi(x) = \lambda\int_{0}^{1}(1 + xt)(\phi (t)\mathrm {d}t, 0\leq x \leq 1\)

Answer
\(\phi(x) = f(x) - \lambda\int R(x,y;\lambda)f(y)\mathrm {d}y\)


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