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We want to evaluate [ I = \int_0^{\infty} e^{-ax} J_0(bx) , dx, \quad a>0, \ b>0, ] where (J_0(x)) is the Bessel function of order zero. The series expansion of (J_0(bx)) is [ J_0(bx) = \sum_{n=0}^{\infty} \frac{(-1)^n _________ ____ ____ _______ _____.
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Check whether satisfies Lipschitz condition
i) on any rectangle and
;
ii) on any strip and
;
iii) on the entire plane.
Using the transformation find the solution of the equation
in terms of Bessel's functions.
Using second order finite Difference method, solve the boundary value problem
If the Fourier cosine transform of f(x) is , then show that
how that between every successive pair of zeros of J0(x) there exists a zero of J1(x).
) Find the displacement u(x, t) of an infinite string using the method of Fourier transform given that the string is initially at rest and that the initial displacement is f(x), .
Find an approximate value of y(1.0) for the initial value problem
y = x - 2y, y(0) = 1
using Milne-Simpson’s method
with the step length . Calculate the starting value using Runge-Kutta fourth order method with the same h.