Mth3006 Cheat Sheet Worked Examples

Worked Examples

1. Fourier Transform (10 marks)

[A1, 10 marks] Calculate the Fourier transform of…

$$f(t)=\{\begin{array}{llc}{e}^{-}t& \text{when}& t\ge 0\\ 0& \text{when}& t<0\end{array}$$

Similar: Question Three, Question Two.

[B1, 10 marks] Calculate, where $a>0$ is a real constant, the Fourier transform of…

$$f(t)=\{\begin{array}{llc}{e}^t& \text{when}& -a<t<a\\ 0& \text{otherwise}& \end{array}$$

Similar: Question Three.

[C1, 10 marks] Calculate the Fourier transform of…

$$f(t)=\{\begin{array}{llc}t& \text{when }& 0\le t\le 1\\ 0& \text{otherwise}& \end{array}$$

Similar: Question Three.

[X2, 10 marks] Calculate the Fourier sine transform of the function $f(t)=\exp (-t)$, using…

$$f_s(\omega )=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(t)\sin (\omega t)dt$$

2. Inverse Laplace Transform (15 marks)

[B2, 15 marks] Use partial fractions to find the inverse Laplace transform of…

$$\tilde{f}(s)=\frac{s^2-15s+41}{(s+2)(s-3{)}^2}$$

Similar: Question Three (part two).

[X3, 15 marks] Show that the inverse Laplace transform of the following can be written in the form $f(t)=a{e}^{3t}+b\cos 2t+c\sin 2t$, and find the values of the constants $a$, $b$, and $c$…

$$\tilde{f}(s)=\frac{5s^2-4s-7}{(s-3)(s^2+4)}$$

[C2, 15 marks] Use the convolution theorem to calculate the inverse Laplace transform of…

$$\tilde{F}(s)=\frac{1}{s^2(s^2-1)}$$

Similar: Question Three (part one).

3. Laplace Transforms (10/15 marks)

[A4, 10 marks] Use Laplace transforms to evaluate the integrals…

$${\int }_0^{\infty }(\sin 3t)e^{-2t}dt\text{and}{\int }_0^{\infty }(t-\cos (3t))e^{-t}dt$$

Similar: Question Four.

[C3, 10 marks] Use Laplace transforms, where $a$ is a real, non-zero constant, to evaluate the integrals…

$${\int }_0^{\infty }{e}^{-3t}{t}^{3}dt\text{and}{\int }_0^{\infty }(\cos (at))e^{-t}dt$$

Similar: Question Four.

[X1, 10 marks] Find the Laplace transform of…

$$f(t)={\int }_0^t{u}^2-u+e^{-u}du$$

[B4, 15 marks] Use Laplace transforms to solve the differential equation, given that $x(0)=1$…

$$\frac{dx(t)}{dt}+3x(t)=e^{-t}$$

Similar: Question One.

4. Method of Characteristics (15 marks)

[A2, 15 marks] Use the method of characteristics to solve the following, subject to the boundary condition $u={\left(1+\frac{1}{x}\right)}^2$ on $y=x$…

$$x\frac{\partial u(x,y)}{\partial x}+(1+y)\frac{\partial u(x,y)}{\partial y}=0$$

Similar: Question Three.

[C4, 15 marks] Use the method of characteristics to solve the following, subject to the boundary condition $u=1$ on $x=0$…

$$\frac{\partial u(x,y)}{\partial x}-x^2\frac{\partial u(x,y)}{\partial y}=2x^{2}y$$

Similar: Question One, Question Two.

5. Change of Variables (10/15 marks)

[B3, 10 marks] Show that making the change of variables $\xi =y+2x$ and $\eta =y-2x$ transforms the differential equation…

$$\frac{{\partial }^{2}u}{\partial x^2}-4\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial x}=0\to 8\frac{{\partial }^{2}u}{\partial \xi \partial \eta }-\frac{\partial u}{\partial \xi }+\frac{\partial u}{\partial \eta }=0$$

Similar: Question Four.

[X4, 15 marks] Transform the following differential equation, where $\gamma$ and $D$ are constants, to a coordinate system given by $\xi =x-\gamma t$ and $\tau =t$…

$$\frac{\partial u}{\partial t}+\gamma \frac{\partial u}{\partial x}=D\frac{{\partial }^{2}u}{\partial x^2}$$

6. Separation of Variables (15 marks)

[A3, 15 marks] Use separation of variables to solve the following, subject to the boundary conditions $u(0,t)=u(2,t)=0$ and $u(x)=3\sin (2\pi x)+2\sin (4\pi x)$…

$$\frac{\partial u(x,t)}{\partial t}=\frac{{\partial }^{2}u(x,t)}{\partial x^2}$$

Similar: Question Three, Question Four.