Fourier Laplace Eb7158

1. **Problem Statement:** (a)(i) Find the Fourier integral representation of the function $$f(x) = \begin{cases} 1 & \text{if } |x| < 1, \\ 0 & \text{if } |x| > 1. \end{cases}$$ (ii) Prove that $$\int_0^\infty \frac{\cos w \sin w}{w} \, dw = \frac{\pi}{4}.$$ (b) Find the Fourier sine transform of $$f(x) = e^{-ax}, \quad a > 0.$$ (c) Find the directional derivative of $$f(x,y) = 3x^3 - 4xy$$ along the parabola $$y = 2x^2 - x + 2$$ at the point $(1,3)$. 2. **Fourier Integral Representation (a)(i):** The Fourier integral representation of a function $f(x)$ is given by $$f(x) = \frac{1}{\pi} \int_0^\infty \left[ A(w) \cos(wx) + B(w) \sin(wx) \right] dw,$$ where $$A(w) = \int_{-\infty}^\infty f(t) \cos(wt) dt, \quad B(w) = \int_{-\infty}^\infty f(t) \sin(wt) dt.$$ Since $f(x)$ is even, $$B(w) = 0,$$ and $$A(w) = 2 \int_0^1 \cos(wt) dt = 2 \left[ \frac{\sin w}{w} \right].$$ So, $$f(x) = \frac{2}{\pi} \int_0^\infty \frac{\sin w}{w} \cos(wx) dw.$$ 3. **Proof of Integral (a)(ii):** Set $x=1$ in the Fourier integral representation: $$f(1) = 1 = \frac{2}{\pi} \int_0^\infty \frac{\sin w}{w} \cos w \, dw.$$ Multiply both sides by $\frac{\pi}{2}$: $$\int_0^\infty \frac{\cos w \sin w}{w} dw = \frac{\pi}{4}.$$ This proves the required integral. 4. **Fourier Sine Transform (b):** The Fourier sine transform of $f(x)$ is $$F_s(w) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \sin(wx) dx.$$ For $f(x) = e^{-ax}$, $$F_s(w) = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-ax} \sin(wx) dx.$$ Using the integral formula $$\int_0^\infty e^{-px} \sin(qx) dx = \frac{q}{p^2 + q^2}, \quad p > 0,$$ we get $$F_s(w) = \sqrt{\frac{2}{\pi}} \frac{w}{a^2 + w^2}.$$ 5. **Directional Derivative (c):** Given $$f(x,y) = 3x^3 - 4xy,$$ and the curve $$y = 2x^2 - x + 2.$$ The directional derivative along the curve at $(1,3)$ is $$D_\mathbf{u} f = \nabla f \cdot \mathbf{u},$$ where $\mathbf{u}$ is the unit tangent vector to the curve. Calculate partial derivatives: $$f_x = 9x^2 - 4y, \quad f_y = -4x.$$ At $(1,3)$, $$f_x(1,3) = 9(1)^2 - 4(3) = 9 - 12 = -3,$$ $$f_y(1,3) = -4(1) = -4.$$ Find $dy/dx$ from the curve: $$\frac{dy}{dx} = 4x - 1,$$ at $x=1$, $$\frac{dy}{dx} = 4(1) - 1 = 3.$$ The tangent vector is $$\mathbf{v} = (1, 3).$$ Unit vector: $$\mathbf{u} = \frac{1}{\sqrt{1^2 + 3^2}} (1, 3) = \frac{1}{\sqrt{10}} (1, 3).$$ Directional derivative: $$D_\mathbf{u} f = (-3, -4) \cdot \frac{1}{\sqrt{10}} (1, 3) = \frac{-3 - 12}{\sqrt{10}} = \frac{-15}{\sqrt{10}} = -\frac{15 \sqrt{10}}{10}.$$ 6. **Laplace Transform IVP (QUESTION FOUR (a)):** Solve $$y'' + 3y' - 4y = 6 e^{2x - 3},$$ with $$y(1.5) = 4, \quad y'(1.5) = 5.$$ Shift variable: let $t = x - 1.5$, then $$y''(t+1.5) + 3 y'(t+1.5) - 4 y(t+1.5) = 6 e^{2(t+1.5) - 3} = 6 e^{2t}.$$ Define $Y(t) = y(t+1.5)$, initial conditions $$Y(0) = 4, \quad Y'(0) = 5.$$ Taking Laplace transform $\mathcal{L}$, $$s^2 Y(s) - s Y(0) - Y'(0) + 3 (s Y(s) - Y(0)) - 4 Y(s) = 6 \frac{1}{s - 2}.$$ Substitute initial values: $$s^2 Y(s) - 4s - 5 + 3s Y(s) - 12 - 4 Y(s) = \frac{6}{s - 2}.$$ Group terms: $$(s^2 + 3s - 4) Y(s) = \frac{6}{s - 2} + 4s + 5 + 12.$$ Simplify right side: $$4s + 17 + \frac{6}{s - 2}.$$ So, $$Y(s) = \frac{4s + 17}{s^2 + 3s - 4} + \frac{6}{(s - 2)(s^2 + 3s - 4)}.$$ Factor denominator: $$s^2 + 3s - 4 = (s + 4)(s - 1).$$ Partial fraction decomposition and inverse Laplace transform yield the solution $Y(t)$, then $$y(x) = Y(x - 1.5).$$ 7. **Non-homogeneous System (QUESTION FOUR (b)):** Solve $$\begin{cases} y_1' = y_1 - 2 y_2 + \cos t, \\ y_2' = -2 y_1 + y_2 - \sin t. \end{cases}$$ Write in matrix form: $$\mathbf{y}' = A \mathbf{y} + \mathbf{g}(t),$$ where $$A = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix}, \quad \mathbf{g}(t) = \begin{pmatrix} \cos t \\ -\sin t \end{pmatrix}.$$ Find homogeneous solution by eigenvalues and eigenvectors of $A$. Solve particular solution by undetermined coefficients or variation of parameters. General solution is sum of homogeneous and particular solutions. **Final answers:** - Fourier integral representation: $$f(x) = \frac{2}{\pi} \int_0^\infty \frac{\sin w}{w} \cos(wx) dw.$$ - Integral evaluation: $$\int_0^\infty \frac{\cos w \sin w}{w} dw = \frac{\pi}{4}.$$ - Fourier sine transform: $$F_s(w) = \sqrt{\frac{2}{\pi}} \frac{w}{a^2 + w^2}.$$ - Directional derivative at $(1,3)$: $$D_\mathbf{u} f = -\frac{15 \sqrt{10}}{10}.$$ - Laplace IVP solution involves inverse transform of $$Y(s) = \frac{4s + 17}{(s + 4)(s - 1)} + \frac{6}{(s - 2)(s + 4)(s - 1)}.$$ - Non-homogeneous system solved by eigen decomposition and particular solution methods.