Solve your IGNOU Doubts

Question:

State whether the following statements True or False? Justify your answers: $5 \times 2 = 10$
a) The function $\|\cdot\|$ defined on $\mathbb{R}^n$ as:
$\|x\| = \sum_{j=1}^{n} |a_j|$ for $x = (a, a_2, \dots, a_n) \in \mathbb{R}^n$
is a norm.

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Question:

) Using fourth order Taylor series method with $h = 0.2$ , solve the initial value problem

$$y' = x + \cos y, \quad y(0) = 0$

upto $x = 1$ .

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Question:

Find an approximate value of y(1.0) for the initial value problem

y = x - 2y, y(0) = 1

using Milne-Simpson’s method

$$y_{n+1} = y_{n-1} + \frac{h}{3} [f_{n+1} + 4f_n + f_{n-1}]$

with the step length $h = 0.2$ . Calculate the starting value using Runge-Kutta fourth order method with the same h.

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Question:

Using standard five point formula, solve the Laplace equation $\nabla^2 u = 0$ in R where R is the square $0 \le x \le 1, 0 \le y \le 1$ 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 an approximate value of y(1.0) for the initial value problem

$y' = x^3 - y^3, y(0) = 1$

using the multiple method

$y_{n+1} = y_n + \frac{h}{3} [7f_n - 2f_{n-1} + f_{n-2}]$

with step length $h = 0.25$ . Calculate the starting values using Runge-Kutta second order method with the same h.

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Question:

Solve the wave equation $u_{tt} = u_{xx}$ with the initial and boundary conditions

$u(x, 0) = 0, u_t(x, 0) = 0, u(0, t) = 0, u(1, t) = 200 \sin \pi t$ .

with $h = k = 0.25$ , using the explicit method upto four time levels.

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Question:

Using second order finite Difference method, solve the boundary value problem

$y'' + 5y' + 4y = 2, y(0) = 0, y(1) = 0, h = 1/2$

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Question:

Solve the heat conduction equation $u_t = u_{xx}$ in the region $R(0 \le x \le 1, t > 0)$ with the initial and boundary conditions $u(x, 0) = 0, u(0, t) = 0, u(1, t) = t$ using Crank-Nicolson method with $h = 0.20$ and $\lambda = 1$ upto two time steps.

Based on the third image provided, here is the transcription of the remaining mathematical problems:

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Question:

Using Runge-Kutta second order method with

(i) $h = 0.2$ , (ii) $h = 0.3$ , solve the initial value problem

$$y' = y^2 \sin x, \quad y(0) = 1.$

Upto $x = 0.6$ . If the 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} d\alpha = \begin{cases} 0 & \text{if } x < 0 \\ \pi / 2 & \text{if } x = 0 \\ \pi e^{-x} & \text{if } x > 0 \end{cases}$

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Question:

) 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), $-\infty < x < \infty$ .

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Question:

If the Fourier cosine transform of f(x) is $\alpha^n e^{-a\alpha}$ , 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 the Laplace transform technique:

$$u_t = 2u_{xx}, \quad 0 < x < 1, \quad t > 0$

$$u(0, t) = 1, \quad u(1, t) = 1, \quad t > 0$

$$u(x, 0) = 1 + \sin \pi x, \quad 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 $p \ge 0, b > 0$ then show that
$$\int_0^b x J_p(k_m x) J_p(k_n x) dx = \begin{cases} 0 & \text{if } m \neq n \\ \frac{b^2}{2} [J_{p+1}(k_n b)]^2 & \text{if } m = n \end{cases}$