Advanced Mathematics For Engineering Students The Essential Toolbox 1st Edition Brent J Lewis, Nihan Onder, E Nihan Onder, Andrew Prudil - Solutions

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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 obtains the specific result
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 \(90 \%\) confidence?
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 method \(\left(x_{0}=0.5, x_{1}=1ight.\), six decimal places \()\).
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 \(\ln 9.4, \ln 10, \ln 10.5, \ln 11.5, \ln 12\). Evaluate the errors by using exact values to four decimal
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 polynomial for interpolation?
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) Simpson's rule \((2 m=4)\); and (iii) the Gaussian integration formula \((n=4)\).(c) Compute the error
The probability \(P\) that a measurement will fall within \(t\) standard deviations is given by \(P=\int_{\mu-t \sigma}^{\mu+t \sigma} \frac{\exp \left(\frac{-(x-\mu)^{2}}{2 \sigma^{2}}ight)}{\sigma \sqrt{2 \pi}} d x\).(a) Show how this probability expression reduces to the normal error integral
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 to four significant digits using the zeros of the Legendre polynomial \(P_{2}(x)\). Using the
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 \(\frac{2}{z^{2}-1}=\frac{1}{z-1}-\frac{1}{z+1}\).(b) Evaluate the integral using Cauchy's integral
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 conditions.
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 upper half-plane of radius \(ho\) centered at the origin.
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 can be listed as pendulum's mass \((m)\), length of the rigid rod \((l)\), pendulum's period
Apply the nondimensionalization technique to the Navier-Stokes equation for the \(x\)-component; the incompressible Navier-Stokes momentum equation is given by\[ho\left[\frac{\partial u}{\partial t}+(u \cdot abla) uight]=-abla P+\mu abla^{2} u+ho g\]where \(ho\) is the density, \(P\) is the
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 hydrostatic stress, where the pressure \(p\) acting to expand the bubble is balanced by the surface
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 \(\lambda\) and the kernel \(K(x, y)\) :(a) \(u(x)=x+\int_{0}^{1} x y u(y) d y\)(b)
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 method.
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 frictional effects, calculate the following:(a) Given that the velocity of the wedge is simply \(\dot{x}\),
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 \(c\) and \(d\) are constants of integration. For this constrained problem, \(y(0)=a, y(1)=b\), 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 \(y_{2}(0)=\) \(-\frac{1}{2}\). Solve this initial value problem by the following three methods:(a) Matrix
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 that this differential equation reduces to \(\frac{d^{2} y}{d t^{2}}+\lambda y=0\).(c) Given the
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 and give the expression for the Legendre polynomial which is a solution to this transformed 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 about \(x=0\) is the geometric series such that \(y_{1}(x)=1+x+x^{2}+\ldots\)
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 the substitution or the generalized form for the solution of the Bessel equation. What is the solution
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 \(a_{m}=2 /\left\{\alpha_{m 0} J_{1}\left(\alpha_{m 0}ight)ight\}\), where \(\alpha_{m 0}\) are the zeros of \(J_{0}\).
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 \(f(x)\) over the interval \(-1 \leq x \leq 1\). What is the value of the complete Fourier-Legendre series of
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 function from \(y_{1}(x)\) using a method of reduction of order.
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 \(z=x^{2}\) and (ii) employing the inspection method for the generalized form of Bessel's equation.
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 differential equation:\[\frac{d}{d x}\left(k A \frac{d T}{d x}ight)-h P\left(T-T_{\infty}ight)=0\]where
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) The Laguerre polynomial \(L_{n}(x)\) is in fact a solution of this differential equation as given by
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 equation reduces to the Bessel differential equation\[\frac{d^{2} u}{d z^{2}}+\frac{1}{z} \frac{d
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 coefficients,(b) variation of parameters,(c) Fourier transforms.
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 function.
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 functions.(b) What is the Laplace transform of \(f(x)\) ?
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 answer to the solution in Problem 2.15(a) and (b)).
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 \(f(0)=\frac{1}{a \sqrt{\pi}}\), where \(a\) is the width of the curve at half the maximum value.(a) Show
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 of this function \(f(x)\) is \(F(w)=\) \(\sqrt{\frac{2}{\pi}}\left[\frac{1}{1+(w a)^{2}}ight]\).(b)
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 method;(b) Cramer's rule;(c) Gauss elimination;(d) Maple software package.
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 \(A=(0,0)\) to \(B=(2,1)\) ?(b) parabolic where \(y=\frac{1}{4} x^{2}\) from \(A=(0,0)\) to \(B=(2,1)\) ?(c)
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 (a), since the force is conservative, any path can be taken to evaluate the line integral.
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 temperature zero, show that the temperature distribution \(u(r, t)\) in the ball can be described by the
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 center of the sphere is finite for \(t>0\), (ii) the surface of the sphere is maintained at a zero
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 \(x\) direction with an average wind speed \(u\), while the material diffuses in the other directions. In the
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 uwuw. The partial differential equation for the temperature distribution u(x,t)u(x,t) for thermal
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 partial differential equation with two independent variables.(a) \(\frac{\partial^{2} u}{\partial
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 \(0^{\circ} \mathrm{C}\). The partial differential equation for the temperature distribution \(u(x,
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 \(10^{\circ} \mathrm{C}\) at \(x=0\) and \(40^{\circ} \mathrm{C}\) at \(x=3\). The partial differential equation
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 Fermi age equation, \(\frac{\partial q}{\partial \tau}=\frac{\partial^{2} q}{\partial x^{2}}\),
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 \(y_{n+1}\).
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 Runge-Kutta method (of fourth order) and solve the problem for \(0 \leq x \leq 1\), with a step size of
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, the bottom and side boundaries each have a temperature of \(100^{\circ} \mathrm{C}\) and the top
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 temperature of \(-10^{\circ} \mathrm{C}\) ), a thin car window with dimensions \(1 \mathrm{~m}\) high by
Consider the temperature distribution \(u(x, t)\) in an insulated bar of length \(L\) as governed by the heat conduction equation \(\frac{\partial u}{\partial t}=\kappa \frac{\partial^{2} u}{\partial x^{2}}\), subject to the boundary conditions \(\left.\frac{\partial u}{\partial x}ight|_{x=0}=0
Explain what methodology could be used to determine if a numerical solution for a given problem is correct and accurate.

Advanced Mathematics For Engineering Students The Essential Toolbox 1st Edition Brent J Lewis, Nihan Onder, E Nihan Onder, Andrew Prudil - Solutions

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