Questions and Answers of Advanced Mathematics

Given the function \(y=a b / c\), where \(a, b\), and \(c\) are independent variables with random errors of \(s_{a}, s_{b}\), and \(s_{c}\), show that using the general error propagation formula one
What is the expression for calculating the standard error of the intercept \(\left(y_{o}ight)\) of a linear regression line? Hint: Use the confidence interval relation to determine this expression.
For the logarithmic expression \(y=\log x\), show that the standard deviation \(s_{y}\) is given by \(s_{y}=0.434\left(s_{x}ight)_{r}\).
A concentration measurement (in ppm) was made consisting of the following values: \(21,15,23,21\), and 24 . Should the anomalously low value be refused based on a \(Q\) critical value at a \(90 \%\)
What is the difference between a \(t\) distribution and a normal distribution as used for confidence interval prediction?
Solve \(x=\cos x\) by:(a) fixed point iteration ( \(x_{0}=1,20\) steps, six significant figures);(b) Newton's method ( \(x_{0}=1\), six decimal places) (sketch the function first);(c) the secant
Consider the following tabular data: \(\ln 9.0=2.1972, \ln 9.5=2.2513\), and \(\ln 11.0=2.3979\).(a) Calculate the Lagrange interpolation polynomial \(p_{2}(x)\) and compute approximations of \(\ln
Find the cubic spline to the given data with \(k_{0}=-2\) and \(k_{2}=-14\). Data: \(f_{0}=\) \(f(-2)=1, f_{1}=f(0)=5, f_{2}=f(2)=17\). What is the advantage of a spline fit compared to a Lagrange
Evaluate the definite integral \(J=\int_{0}^{1} \frac{d x}{1+x^{2}}\) using:(a) an exact analytic formula obtained from calculus;(b) approximate methods: (i) the trapezoidal rule \((n=4)\); (ii)
The probability \(P\) that a measurement will fall within \(t\) standard deviations is given by \(P=\int_{\mu-t \sigma}^{\mu+t \sigma} \frac{\exp \left(\frac{-(x-\mu)^{2}}{2 \sigma^{2}}ight)}{\sigma
The sine integral is defined as \(\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin u}{u} d u\). Evaluate the quantity \(\operatorname{Si}(1)\) (that is, for \(x=1\) ) using a Gaussian integration formula
Expand the function \(f(z)=\frac{1}{(z+1)(z+3)}\) in a Laurent series, which is valid for \(1
Consider the integral \(\oint_{C} \frac{2 d z}{z^{2}-1}\), where \(C\) is a circle of radius \(1 / 2\) and centered at 1 , and positively oriented.(a) Using a partial fraction method, show that
Given the function \(u(x, y)=x^{2}-y^{2}\).(a) Prove that this equation satisfies the Laplace equation.(b) Find the function \(v(x, y)\) such that \(f(z)=u+i v\) is an analytic function.
Show that \(f(z)=\sin z\) is analytic in the following steps:(a) Find the real and imaginary parts of \(f(z)\).(b) Show that \(u(x, y)\) and \(v(x, y)\) from part (a) satisfy the Cauchy-Riemann
Evaluate the integral \(\oint_{C} z^{2} d z\) along the path shown in Fig. 10.9. FIGURE 10.9 -1 0 C +1
Evaluate the integral \(\oint_{C} \frac{d z}{\left(z^{2}-a^{2}ight)} d z\), where \(C\) is a circle of radius \(b\) centered at the origin with \(b>a\).
Evaluate the definite integral \(\int_{-\infty}^{+\infty} \frac{d x}{\left(1+x^{2}ight)}\), where the contour runs along the \(x\) axis from \(-ho\) to \(+ho\), and then closes by a semicircle in the
Determine the variables on which a pendulum's amplitude (or sometimes called the pendulum position) depends (see Fig. 11.2). Use the Buckingham \(\pi\) theorem; the relevant dimensional variables
Apply the nondimensionalization technique to the Navier-Stokes equation for the \(x\)-component; the incompressible Navier-Stokes momentum equation is given by\[ho\left[\frac{\partial u}{\partial
As shown in Lewis et al. (2017), fuel swelling in nuclear fuel is caused by the presence of fission gases in small bubbles. Consider a gas bubble of radius \(R\) embedded in a solid medium with no
Solve the following integral equation using a Laplace transform method:\[y(t)=\sin t+\int_{0}^{t} y(\tau) \sin (t-\tau) d \tau\]
Classify each of the following integral equations as a Fredholm- or Volterratype integral equation, as linear or nonlinear, and as homogeneous or nonhomogeneous, and identify the parameter
Convert the following Volterra integral equation into an initial value problem: \(u(x)=x+\int_{0}^{x}(y-x) u(y) d y\).
Convert the following initial value problem into a Volterra integral equation: \(u^{\prime \prime}(x)+u(x)=\cos x\), subject to the initial conditions \(u(0)=0\) and \(u^{\prime}(0)=1\).
Convert the following boundary value problem into a Fredholm integral equation: \(y^{\prime \prime}(x)+y(x)=x\), for \(x \in(0,1)\), subject to the boundary conditions \(y(0)=1\) and \(y(1)=0\).
Given the Fredholm integral equation \(u(x)=x+\lambda \int_{0}^{1}(x y) u(y) d y\), solve this equation using (i) the successive approximation method, (ii) a Neumann series, and (iii) the resolvent
Given the Volterra integral equation \(u(x)=1+\int_{0}^{x}(x-y) u(y) d y\), solve this equation using the successive approximation method.
Consider a small block of mass \(m\) sliding a distance \(s\) down a moving wedge of mass \(M\), as shown in Fig. 14.8. The wedge is also sliding a distance \(x\) along the floor. Neglecting
Show that the shortest curve that has an area \(A\) below it is a circular arc,\[(\lambda x-c)^{2}+(\lambda y-d)^{2}=1,\]as shown in Fig. 14.9. Here \(\lambda\) is a Lagrange multiplier constant and
Consider the nonhomogeneous system of first-order, linear differential equations \(y_{1}^{\prime}=y_{2}+\cosh t\) and \(y_{2}^{\prime}=y_{1}\) with boundary conditions \(y_{1}(0)=0\) and
Consider the following problems:(a) State the differential equation for the Sturm-Liouville problem where \(r(x)=\) \(x, p(x)=x^{-1}\), and \(q(x)=0\).(b) Using the transformation \(x=e^{t}\), show
Using the transformation \(x=\cosh t\), show that the differential equation \(\frac{d^{2} y}{d t^{2}}+\) \(\operatorname{coth} t \frac{d y}{d t}-20 y=0\) reduces to Legendre's differential equation
Consider the hypergeometric equation \(x(1-x) y^{\prime \prime}+[b-(2+b) x] y^{\prime}-b y=0\), where \(b\) is a constant larger than unity. Using a Frobenius method, show that the series solution
Find a general solution, in terms of Bessel functions, for the differential equation \(4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x-n^{2}ight) y=0\), where \(n\) is an integer, by using either
Using \(J_{o}\) for a series expansion, show that the coefficients of the Fourier-Bessel series for \(f(x)=1\) over the interval \(0 \leq x \leq 1\) are given as \(a_{m}=2 /\left\{\alpha_{m 0}
Consider the function \(f(x)= \begin{cases}0 & \text { if }-1 \leq x \leq 0, \\ 2 & \text { if } 0 \leq x \leq 1\end{cases}\)Calculate the first two terms of the Fourier-Legendre series of \(f(x)\)
Consider the differential equation \(16 x^{2} y^{\prime \prime}+3 y=0\).(a) Calculate the two basis functions using a Frobenius method and give the general solution.(b) Derive the second basis
Give the Legendre polynomial \(P_{n}(x)\) which is a solution to the differential equation \(\frac{1}{2}\left(1-x^{2}ight) y^{\prime \prime}-x y^{\prime}+3 y=0\).
Find a general solution, in terms of Bessel functions, for the differential equation \(x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-\left(1+4 x^{4}ight) y=0\) by (i) using the transformation
The steady-state temperature distribution for a fin of cross-sectional area \(A\), constant perimeter \(P\), constant conductivity \(k\), and length \(L\) can be determined from the following
Given the periodic function \(f(x)=x\) for \(-2
Given is the differential equation \(x y^{\prime \prime}+(1-x) y^{\prime}+n y=0\).(a) Determine one of the basis solutions for this differential equation, when \(n=2\), using a Frobenius method.(b)
Consider the differential equation \(\frac{d^{2} y}{d x^{2}}-x y=0\). Using the transformations \(y=u \sqrt{x}\) and \(\frac{2}{3} i x^{3 / 2}=z\), where \(i=\sqrt{-1}\), show that this differential
Determine the particular solution for the nonhomogeneous, second-order, ordinary differential equation \(y^{\prime \prime}+y^{\prime}-2 y=4 \sin 2 x\) using the following methods:(a) undetermined
Find \(\mathscr{L}^{-1}\left\{\frac{s}{\left(s^{2}-a^{2}ight)^{2}}ight\}\) using the property for differentiation of a transform.
Find \(\mathscr{L}\left\{\frac{\cos a t-\cos b t}{t}ight\}\) using the property for integration of a transform.
Find \(\mathscr{L}^{-1}\left\{\frac{1}{s^{4}-1}ight\}\) using the convolution theorem.
Find the inverse Laplace transform \(\mathscr{L}^{-1}\left\{\frac{e^{-\pi s} s}{\left(s^{2}+1ight)}ight\}\) using the second shifting theorem, and sketch the resultant function.
Find the inverse Laplace transform \(\mathscr{L}^{-1}\left\{\frac{1}{\left(s^{2}-1ight)}ight\}\) using the property for the differentiation of a transform.
Find the inverse Laplace transform \(\mathscr{L}^{-1}\left\{\left(\frac{\Gamma(2)}{s^{2}}ight)\left(\frac{s}{s^{2}+1}ight)ight\}\) using the convolution theorem, where \(\Gamma\) is the gamma
Consider the function \(f(x)= \begin{cases}0 & \text { if }-1 \leq x \leq 0, \\ 2 & \text { if } 0 \leq x \leq 1 .\end{cases}\)(a) Sketch this function and write it in terms of Heaviside step
Find the Fourier transform \(\mathscr{F}\{f(x)\}\) by direct integration, where \(f(x)\) is given by \(f(x)= \begin{cases}e^{x} & \text { if } x0\end{cases}\)
Show that the Fourier sine transform pair as given in mathematical handbooks (for example, Spiegel, 1973),\[f(x)=x^{a-1} \quad(0
Determine the particular solution for the nonhomogeneous, second-order, ordinary differential equation \(y^{\prime \prime}+y^{\prime}-2 y=4 \sin 2 x\) using a Fourier transform method (compare your
Show that the normal distribution reduces to \(f(x)=\frac{e^{-(x / a)^{2}}}{a \sqrt{\pi}}\) if it is centered at the origin, where \(a=\sqrt{2} \sigma\). The maximum of this function occurs at
Given the function \(f(x)=\frac{1}{a} e^{-|x| / a}= \begin{cases}\frac{1}{a} e^{-x / a} & \text { for } x>0, \\ \frac{1}{a} e^{x / a} & \text { for } x(a) Show that the Fourier transform
Given the following system of equations:\[\begin{aligned}& 3 y+2 x=z+1, \\& 3 x+2 z=8-5 y, \\& 3 z-1=x-2 y\end{aligned}\]solve for \(x, y\), and \(z\) using the following methods:(a) Inverse
What is the work done if \(\vec{F}=x y \hat{i}-y^{2} \hat{j}\) and \(\overrightarrow{d r}=\hat{i} d x+\hat{j} d y\) assuming the path is:(a) a straight line where \(y=\frac{1}{2} x\) from \(A=(0,0)\)
Let the force \(\vec{F}=\left(2 x y-z^{3}ight) \hat{i}+x^{2} \hat{j}-\left(3 x z^{2}+1ight) \hat{k}\).(a) Show that \(\vec{F}\) is conservative.(b) Find \(\phi(x, y, z)-\phi(0,0,0)\). Note that from
Consider a solid ball of radius \(R\), with a constant thermal diffusivity \(\kappa\). Initially, the ball has a uniform temperature \(u_{o}\). If this ball is dropped into an ice-water bath of
Consider the radial diffusion of material in a sphere of radius \(a\), with a concentration distribution \(C(r, t)\) and constant diffusivity \(D\). It is assumed that: (i) the concentration at the
A standard model for describing the dispersion of chemical pollutants in the atmosphere is the Gaussian plume model. In this model it is assumed that the wind carries the pollutant in the \(x\)
Consider heat conduction in a semiinfinite slab, where the slab is initially (t=0)(t=0) at a constant temperature umum and the end of the slab (at x=0x=0 ) is maintained at the constant temperature
Give a brief physical interpretation of the following partial differential equations. Classify each equation as elliptic, hyperbolic, or parabolic by comparing it to the general form of a linear
Consider the heat conduction in a thin insulated bar of length \(3 \mathrm{~m}\) where the initial temperature at \(t=0\) is \(f(x)=15-10 x^{\circ} \mathrm{C}\) and the ends of the bar are kept at
Consider the heat conduction in a thin insulated bar of length \(3 \mathrm{~m}\) where the initial temperature at \(t=0\) is \(25^{\circ} \mathrm{C}\) and the ends of the bar are kept at \(10^{\circ}
The slowing down of neutrons in the moderator of a nuclear reactor can be described by the slowing down density \(q\) (neutrons \(\mathrm{m}^{-3} \mathrm{~s}^{-1}\) ) in accordance with the so-called
Explain the advantages and disadvantages of the Adam-Moulton, Runge-Kutta, and Euler methods for the numerical solution of first-order ordinary differential equations.
Consider the first-order ordinary differential equation \(y^{\prime}=\frac{d y}{d x}=f(x, y)\). Derive the predictor and corrector equations for the Heun method by using a truncated Taylor series for
Given the initial value problem \(y^{\prime}=y-y^{2}, y(0)=0.5\) :(a) Solve the problem exactly using standard analytical methods for first-order ordinary differential equations.(b) Apply the
Using the Maple software package:(a) Consider the square plate in Example 6.2 . 1 for a mesh spacing of \(h=4 \mathrm{~cm}\) (that is, \(h=a / n\), where \(a=12 \mathrm{~cm}\) and \(n=3\) ). However,
Write the steady-state heat conduction equation (that is, the Laplace equation) for a Cartesian coordinate system for two spatial dimensions. During the winter (in which the air has an ambient
Consider the temperature distribution \(u(x, t)\) in an insulated bar of length \(L\) as governed by the heat conduction equation \(\frac{\partial u}{\partial t}=\kappa \frac{\partial^{2} u}{\partial
Explain what methodology could be used to determine if a numerical solution for a given problem is correct and accurate.