Solve your IGNOU Doubts

Question:

Using fourth order Taylor series method with $h=0.2$ , solve Initial value problem $y' = x + cos y , \: y(0) = 0$ upto $x=1$ .

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Question:

Find approximate value of $y(1.0)$ for the initial value problem $\begin{align*} y' =x-2y, y(0)=1 \end{align*}$ using Milne-Simpson's method $y_{n+1} = y_{n-1} + \frac{h}{3}[f_{n+1} + 4f_{n} - f_{n-1}]$
with $h=0.2$ . Calculate starting value using Runge-Kutta fourth order method with the same h.

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Question:

Using standard five-point formula, solve Laplace equation $\nabla^2u=0$ in R where R is the square $0 \leq x \leq 1, 0 \leq y \leq 1$ subject to the boundary conditions $u(x,y) = x^2 - y^2$ on $x=0 , y=0, y=1$
and $3u + 2\frac{\partial u}{\partial x} = x^2 + y^2$ on $x=1$ . Assume $h=k=1/2$ .

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Question:

Find approximate value of $y(0,1)$ for initial value problem
$y'=x^3-y^3 , y(0)=1$ using multiple method

$\begin{equation} y_{n+1} = y_n + \frac{h}{3}(7f_n - 2f_{n-1} + f_{n-2}) \end{equation}$

with $h=0.2$ . Calculate the starting values using Runge-Kutta second order method with the same h.

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Question:

Solve wave equation $u_{tt}=u_{xx}$ with
$u(x,0) = 0 , u_t(x,0)=0, u(0,t)=0 , u(1,t) = 100 sin(\pi t)$ with $k=h=0.25$ , using explicit method upto 4 time levels.

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Question:

Using second order finite difference method, solve the boundary value problem $y'' + 5y' + 4y= 1$ , $y(0) = 0$ , $y(1) = 0$ , $h=1/4$

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Question:

Solve heat equation $u_{tt}=u_{xx}$ in $R(0 \leq x \leq 1, t > 0)$ with conditions $u(x,0)=0, u(0,t)=0, u(1,t)=t$ using Crank-Nicolson method with $h=0.25$ , $\lambda=1$ upto two time steps.

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Question:

Using Runge-Kutta 2nd order method with

(i) $h=0.1$ , (ii) $h=0.2$ , solve the initial value problem
$\begin{align*} y'&=y^2\sin x , y(0) =1 \end{align*}$ upto $x=0.4$ . If exact solution is $y = sec x$ , obtain the error.

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Question:

Using Fourier integral representation show that $\int_{0}^{\infty}\frac{cos(\alpha x) + \alpha sin(\alpha x)}{1 + \alpha^2} = \left\{\begin{matrix} 0 & if \: x<0\\ \pi/2 & if \: x=0 \\ \pi e^{-x} & if \: x>0 \end{matrix}\right.$

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Question:

Find the displacement $u(x,t)$ of an infinite string using Fourier transform method given that the string is initially at rest and the initial displacement is $f(x)$ , $-\infty < x < \infty$ .

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Question:

If the Fourier cosine transform of $f(x)$ is $\alpha^ne^{-a\alpha}$ , then show that

$f(x) = \frac{2}{\pi}\frac{n!cos(n+1)\theta)}{(a^2+x^2)^{\frac{n+1}{2}}}$

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Question:

If the Fourier cosine transform of $f(x)$ is $\alpha^ne^{-a\alpha}$ , then show that

$f(x) = \frac{2}{\pi}\frac{n!cos(n+1)\theta)}{(a^2+x^2)^{\frac{n+1}{2}}}$

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Question:

Solve the following IBVP using Laplace transform technique:
$u_t = u_{xx} , 0<x<1, t>0$
$u(0,t) = 1, u(1,t) = 1, t>0$
$u(x,0) = 1 + sin \pi x, 0<x<1$

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Question:

If $k_m$ and $k_n$ are distinct roots of Bessel function $J_p(kb) = 0$ with $p\geq0$ , $b>0$ then show that $\int_{0}^{b}x J_p(k_mx)J_p(k_nx)dx = \left\{\begin{matrix} 0 & if m\neq n\\ \frac{b^2}{2}[J_{p+1}(k_nb)] & if \: m = n \end{matrix}\right.$