IGNOU MMT-007 Solved Assignment 2023 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-007 Assignment Question Paper 2023

Course Code: MMT-007

Assignment Code: MMT-007/TMA/2023

Maximum Marks: 100

1. State whether the following statements are True or False. Justify your answer with the help of a short proof or a counter-example:

$$(5 \times 2=10)$$

i) Initial value problem:

$\frac{d y}{d x}=\frac{y-1}{x}$

$$y(0)=1$$ has a unique solution.

ii) The second order Runge-Kutta method when applied to IVP $$y^{\prime}=-100 y, y(0)=1$$ will produce stable results for $$0<h<\frac{1}{50}$$.

iii) If Fourier cosine transform of $$f(x)$$ is:

$F_{c}(n)=\frac{\cos \left(\frac{2 n \pi}{3}\right)}{(2 n+1)^{2}}$

where $$0 \leq x \leq 1$$, then:

$f(x)=1+2 \sum_{n=1}^{\infty} \frac{\cos \left(\frac{2 n \pi}{3}\right)}{(2 n+1)^{2}} \cos n \pi x .$

iv) For the differential equation $$x^{2}(x-4)^{2} y^{\prime \prime}(x)+3 x y^{\prime}(x)-(x-4) y=0, x=0$$, is a regular singular point and $$x=4$$, is an irregular singular point.

2. a) Find the power series solution of the equation:

$x^{2} \frac{d^{2} y}{d x^{2}}+4 x \frac{d y}{d x}+\left(x^{2}+2\right) y=0$

b) Construct Green’s function for the following boundary value problem:

$\frac{d^{2} y}{d x^{2}}+9 y=0$

with $$y(0)=y(1)=0$$

3. a) Find the solution of $$\nabla^{2} u=0$$ in $$R$$ subject to $$R$$ : triangle $$0 \leq x \leq 1,0 \leq y \leq 1$$, $$0 \leq x+y \leq 1$$ and $$u(x, y)=x^{2}-y^{2}$$ on the boundary of the triangle. Assume $$h=\frac{1}{4}$$ and use five-point formula.

b) Show that the method:

$y_{i+1}=\frac{4}{3} y_{i}-\frac{1}{3} y_{i-1}+\frac{2 h}{3} y_{i+1}^{\prime}$

is absolutely stable when applied to the equation $$y^{\prime}=\lambda y, \lambda<0$$.

c) Evaluate $$L^{-1}\left\{\frac{1}{(s+1)\left(s^{2}+1\right)}\right\}$$, using convolution theorem.

4. a) Heat conduction equation $$u_{t}=u_{x x}$$ is approximated by the method:

$u_{m}^{n+1}-u_{m}^{n-1}=\frac{2 k}{h^{2}} \delta_{k}^{2} u_{m}^{n}$

Find the order of the method and investigate the stability of this method using Von Neumann method.

b) Using second order finite differences method with $$h=\frac{1}{2}$$, obtain the system of equations for $$y_{0}, y_{1}$$ and $$y_{2}$$ for solving the boundary value problem:

$y^{\prime \prime}-5 y^{\prime}+6 y=3$

with $$y(0)-y^{\prime}(0)=-1$$ and $$y(1)+y^{\prime}(1)=1$$

5. a) Given $$\frac{d y}{d x}=x-y^{2}, y(0.2)=(0.02)$$. Find $$y(0.4)$$ by using modified Euler’s method, correct to two decimal places, taking $$h=0.2$$.

b) Determine an appropriate Green’s function either by using the method of variation of parameters or otherwise, for the following boundary value problem:

$-\left(y^{\prime \prime}(x)-y(x)\right)=\frac{2}{1+e^{-x}}, y(0)=y(1)=0 \text {. }$

6. a) Find the solution of the initial boundary value problem $$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, 0 \leq x \leq 1$$ $$u(x, 0)=\sin \pi x, 0 \leq x \leq 1 \frac{\partial u}{\partial t}(x, 0)=0, u(0, t)=u(1, t)=0, t>0$$ by using second order explicit method with $$h=\frac{1}{4}, r=\frac{1}{3}$$. Integrate for one time step.

b) Prove that

$\frac{d}{d x}\left(J_{n}^{2}(x)\right)=\frac{x}{2 n}\left[J_{n-1}^{2}(x)-J_{n+1}^{2}(x)\right]$

$$sin\left(2\theta \right)=2\:sin\:\theta \:cos\:\theta$$

MMT-007 Sample Solution 2023

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$$sin^2\left(\frac{\theta }{2}\right)=\frac{1-cos\:\theta }{2}$$

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