MMT-007 Solved Assignment 2023

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

about its singular point.

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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