- Explain the mathematical modelin the context of Eq. (1.1.1). = y dx (1.1.15)
- Solve the unsteady diffusion equationsubject to boundary conditions c(0, t) = 1 and ∂c/∂x = 0 at x = 1. The initial condition is given by the piecewise functionYou should also provide the
- Show Eq. (2.11.9) is true by directly computingand showing that the entries in Eq. (2.11.10) are identical to those in Eq. (2.11.7). LU . [ 1 m2,1 m3,1 0 0 1
- Solve the following system of equations by LU decomposition: 212 -2 0 4 2 40 1 2 -4 0 4 1 XI X2 X3 X4 = 1 13 2 -20 (2.11.27)
- Solveusing successive relaxation with a parameter w = 3/2 and an initial guess of x1 = 0, x2 = 0, x3 = 0. 324 130 24 1 6 CHO X2 3 X3
- Do one iteration of successive relaxation with w = 3/2 using the system in Problem 2.58 and an initial guess x = [−0.9 0.1 2.2 1.1].Data from Problem 2.58Perform the first two iterations of
- Answer the following questions about this program:(a) What mathematical problem is solved by this program?(b) What numerical method is used to solve the problem?(c) What does the variable n
- Determine the value of y such that [3,y] is an eigenvector of the matrixfor the eigenvalue λ = 5. = [42] A =
- Use bracketing for Eq. (3.1.16) using the initial guesses a = −1.51 and b = 0.01 with the initial step size of Δx = 0.1. f(x)=x²-1=0 (3.1.16)
- Repeat Example 3.5 using Newton’s method.Data from Example 3.5Use Picard’s method to find the roots offor the initial guesses −0.1, 0.6, 1.99, and 2.01. f(x)=x²-3x+2=0 (3.3.2)
- Consider the roots of the equation(a) Write Newton’s method for computing x(k+1) from the previous value x(k) for this problem.(b) Simplify your result for large x(k).(c) Assume you have computed
- Investigate the behavior of Newton’s method for the previous example, Eq. (3.3.2), for initial guesses near x = 1.5. f(x)=x²-3x+2=0 (3.3.2)
- Consider the code below that implements Newton’s method and answer the following questions.(a) What is the largest real root of the function studied in this problem?(b) What is the criterion for
- What mathematical problem is solved by the following MATLAB code? 1 function x = problem3_8 2 itmax = 100; tol = 0;, x0 = 1; 3 for k=1:itmax 4 5 6 8 9 x=x-f (x)/fprime (x); r=f(x); if abs (r)
- Use contraction mapping to explain the behavior of the solution to Example 3.5.Data from Example 3.5Use Picard’s method to find the roots offor the initial guesses −0.1, 0.6, 1.99, and 2.01.
- Demonstrate that Newton’s method exhibits quadratic convergence for the problem in Example 3.7.Data from Example 3.7Repeat Example 3.5 using Newton’s method.Data from Example 3.5Use Picard’s
- Answer the following question about this MATLAB function:I get the output x = 0.3333 after one iteration, which is not the correct answer. What is required to fix this code? 1 2 3 4 5 6 7 8 function
- Use Picard’s method to find a root of the system of equationsusing the initial guesses x(0)1 = 0 and x(0)2 = 0. f₁(x₁, x₂) = ex1 - x2 f2(x₁, x₂) = x₁ + x² – 3x2 - (3.7.6) (3.7.7)
- Now try to solve the equations from Example 3.11 using the initial guesses x(0)1 = 0 and x(0)2 = 0, but write the equations in the opposite order,Data from Example 3.11Use Picard’s method to find a
- If I am using Newton’s method and I am near the root with a value |f (x(5))| = 10−2, what would I expect to have for |f (x(7))|? You only need an approximate answer.
- What is the Jacobian needed for the Newton–Raphson solution of the system of equationswith the unknown vector listed as R₁ = x² - y² = 0 R₂ = xy-3=0
- Solve the system of Eqs. (3.7.6) and (3.7.7) using Newton’s method with an initial guess x(0)1 = 0 and x(0)2 = 0. fi(x₁, x₂) = ex1 - x₂ f2(x₁, x₂) = x₁ + x² - 3x2 (3.7.6) (3.7.7)
- Show that Newton’s method for f (x) = x3 does not exhibit quadratic convergence.
- Solve the same problem as Example 3.13 with the equations in the opposite order.Data from Example 3.13Solve the system of Eqs. (3.7.6) and (3.7.7) using Newton’s method with an initial guess x(0)1
- Consider the following nonlinear vector function:Show the form of all the equations needed to perform the Newton -Raphson iteration scheme for solving F(x) = 0. F(x):
- Show that the solution in Example 3.13 converges quadratically.Data from Example 3.13Solve the system of Eqs. (3.7.6) and (3.7.7) using Newton’s method with an initial guess x(0)1 = 0 and x(0)2 =
- Consider the nonlinear system of equations(a) What are the roots for this problem?(b) What is the Jacobian if we write our unknown vector as [x,y]?(c) When I try to solve this with Newton–Raphson,
- This problem deals with the Newton–Raphson solution to the system of equationswith the unknown vectorIf you give this problem the initial guessand get the vectorafter one iteration, what is the
- Find the root at x = 0, y = 0, and z = 1 forusing Newton–Raphson. x² + y² = 0 e-ty ely = Z z³ = 1 (3.7.37) (3.7.38) (3.7.39)
- Consider a system of i = 1, 2, . . . , n residuals of the formwhere aij and bi are constants. Show that applying Newton–Raphson to this system of equations will converge to a solution in one
- Solve Eq. (3.5.11) using fzero. sin λ- λ cos λ = 0 (3.5.11)
- Use explicit Euler to compute the concentration profile for a second-order reaction in batch reactor, with an initial reactant concentration of c0 = 3 mole/liter and a reaction rate constant k = 2
- Compare the exact solution, the result from one time step of forward Euler, and the result of one time step for backward Euler for a step size of h = 0.01, h = 0.1, and h = 1 for the differential
- Use implicit Euler to integratesubject to the initial condition y(0) = 1. dy =x²y¹/2 dx (4.2.32)
- Perform one time step of the predictor–corrector method with h = 0.01 for the differential equation dy dx = cos(x) y2 sin(πx); y(1)=10 -
- Write a program to solveusing implicit Euler. dy dt = y¹/2, y(0) = 2 (4.2.39)
- Perform one time step of RK4 with h = 0.1 for the differential equation dy dx In(xy); y(1)=2
- Use predictor–corrector to integratesubject to the initial condition y(0) = 1. dy dx =x²y¹/2 (4.2.53)
- Imagine that I have integrated the autonomous IVPusing some method and at step n I have a value yn. Under what conditions will explicit Euler and predictor–corrector yield the same result for yn+1?
- Write out the equations required to integrateusing RK4. dy dt = y² +t, y(0) = 1 (4.2.85)
- Use RK4 to integratesubject to the initial condition y(0) = 1. dy = xzyl/2 dx (4.2.91)
- Consider the differential equation y′ = y2 subject to y(0) = 2.(a) What is the value of y from the predictor–corrector scheme after a single step h = 1/4?(b) It is easy to show that the exact
- Consider the following nonlinear ordinary differential equationwith y(0) = 1. Show the exact form of the equations needed to calculate yi+1, with an integration step equal to h, using the:(a)
- Write the equations required to solveby explicit Euler. dy1 dt dy2 dt = y₂e t 2e -YI (4.6.12) (4.6.13)
- Consider the following nonlinear ordinary differential equationwith y(0) = 2. Show the exact form of the equations needed to calculate yi+1, with an integration step equal to h, using the:(a)
- Set up a forward Euler scheme to compute the concentration for a batch reactor with k₁ A+B=C k_1 k2 B+C=D k_2 (4.6.18) (4.6.19)
- Solve the IVPwith y(0) = 0 using ode45. dy dt = e~! (4.10.1)
- Perform a local stability analysis for the differential equationWhat is the step size that you will need at a particular time t and corresponding value ys if you try to integrate this problem with a
- Solve the system of IVPswith y1(0) = y2(0) = 1 using ode45. dy1 dt dy2 dt = y₁y₂ = y² - y² (4.10.2) (4.10.3)
- Use the method of lines to construct a numerical solution to the onedimensional unsteady heat conduction equationsubject to the boundary conditionsas well as an initial condition, aT at = Q а
- Write a MATLAB program that uses centered finite differences to solve the onedimensional diffusion equationfor a spatially dependent diffusion coefficient D(x) = D0(1 − ax) and the boundary
- Let’s return to Example 7.1 and use explicit Euler to integrate the system of equations with α = 1 on the domain x ∈ [0, 1].Example 7.1Use the method of lines to construct a numerical solution
- How many non-zero entries are in the matrix required to solve the two-dimensional diffusion equation ∇2T = 0 with n nodes in each direction and Dirichlet conditions on the boundaries? In the
- Write a MATLAB program that uses implicit Euler and centered finite differences to solve the diffusion–reaction equationsubject to c(0, t) = 1, c(1, t) = 1, and c(x, 0) = 1. Әс at а2с ах2 =
- If it takes 2 seconds to solve a two-dimensional, diffusion equation ∇2T = 0 with 20 nodes in x and y using banded Gauss elimination, how long does it take for 40 nodes in x and y?
- Show that Eq. (8.2.27) reduces to Eq. (8.2.20). f[x₁, xj, xk] = f[xi, xj] - f[xj, xk] Xi – Xk (8.2.27)
- Consider the solution to the unsteady diffusion equationby the method of lines and centered finite differences with a spacing x. The initial condition is c(x, 0) = 0 and the left boundary condition
- Compute the third divided difference using the recursion relationship Eq. (8.2.28). f[xn, xn-1,...,x₁, xo] = = f[xn, xn-1,...,x2, X₁] - f[xn-1,..., X1, Xo] Xn - XO (8.2.28)
- How many points do you need to construct a cubic interpolation?
- Derive the fourth-order finite difference Δ4f.
- Determine a cubic interpolation to tan(x) using the interpolation points x = 0, π/3, π/6, and π/4.
- Simplify Eq. (8.2.26) for equally spaced data. f[xi, xj] = f(xi) - f(xj) x₁ - xj (8.2.26)
- Derive the truncation error for a sixth-order Newton–Coates integration formula.
- Simplify Eq. (8.2.32) for evenly spaced data. f[x2, x1, xo]: = [f(x₂) - f(x₁) X2 - X1 2]: f(x₁)-f(xo) X1 - XO 1 X2 - XO (8.2.32)
- Use Newton’s divided difference interpolation to find erf(0.5) based on the data provided in the following table. You need to go up to the fourth divided difference. Compare your result with that
- If I want to compute the integralusing two point Gauss quadrature, at what values of x do you evaluate the function xcos πx? So xсos лx dx
- Answer the following questions about this code:(a) What mathematical problem is being solved?(b) What numerical method is used to solve the problem?(c) What is the significance of the value of n on
- Simply Eq. (8.2.33) for evenly spaced data. f[X3, X2, X1, Xo] f[x3, x2, x1] - f[x2, X1, Xo] X3 - XO (8.2.33)
- To test our claim about the accuracy of these methods, use Program 8.4 to calculate the integralwhich we know has the value I = 1 for any positive integer m.Program 8.4We can write a nice compact
- Assume you have n+1 data points (xi, yi), i = 0, . . . , n, where x is the independent variable and y is the dependent variable. You suspect that these data come from some unknown function of the
- Use the multiple trapezoidal rule to determine how the error in the integraldepends on the number of trapezoids. = 1/2 соs л xdx (8.4.8)
- Compute the first four polynomial interpolations (linear, quadratic, etc.) for the function f (x) = 1 − exp(−x) with evenly spaced data on the interval x ∈ [0, 3]. Make a plot of the
- Repeat Example 8.7 for Gauss quadrature using Program 8.8.Example 8.8Use the multiple trapezoidal rule to determine how the error in the integraldepends on the number of trapezoids.Example 8.7To test
- Write a program that uses multiple trapezoidal rule to compute the integralYour program should automatically generate a plot of I versus the number n of trapezoids used to compute the answer up to a
- Use Romberg integration to compute the integral from Example 8.8.Example 8.8Use the multiple trapezoidal rule to determine how the error in the integraldepends on the number of trapezoids.
- Write a MATLAB program that computes the integralusing the following methods:(a) Trapezoidal rule(b) Simpson’s 1/3 rule(c) Simpson’s 3/8 rule(d) Boole’s rule(e) Fifth-order Newton–Coates(f)
- Write a MATLAB program that computes the integralusing the following methods:(a) Trapezoidal rule(b) Simpson’s 1/3 rule(c) Simpson’s 3/8 rule(d) Boole’s rule(e) Fifth-order Newton–Coates(f)
- Use integral to evaluate the integral I 1 = [₁ e ²³ dx (8.7.1)
- Use trap to find the integral using the tabulated data in Table 8.5. Table 8.5 Tabulated data for Example 8.12. x 0.0000 0.2500 0.5000 0.7500 y 1.0000 0.9394 0.7788 0.5698 1.0000 0.3679
- An important time parameter in reaction engineering is the reactor space time τ , the time needed to process one reactor volume of fluid. For an ideal plug-flow reactor (PFR), a mass balance gives a
- In reaction engineering, the average residence time ¯t is the amount of time an element of fluid spends in a reactor and is related to the amount of substance present in the system. The easiest
- Breakthrough curves describe the concentration of the solution eluted from an adsorption column over time. The efficiency of a column is calculated from breakthrough curves aswhere Q is the
- The kinetic theory of adsorption of polymers under shear flow near a plane wall leads to the following expression for the steady state concentration C(z) in the wallnormal direction (z-axis)where
- In thermodynamics, you learned that the saturation pressure for a one component vapor–liquid equilibrium can be computed using an integral construction from the phase diagram, where the area
- Derive the equations for the coefficients and function evaluation points that you would need for a n-point Gauss quadrature that is exact for 2n − 1-order polynomials. Write aMATLAB program that
- Use LU decomposition to solve the system of equations I 3 1 1 -1 1 0 0 -01 | 0 X₁ X2 X3 X4 || = 11 -3 17 5 0 2 6 ㅎㅎㅎㅎ
- You are solving a problem using LU decomposition. At the end of the solution, you have come up with the upper triangular matrixand the lower triangular matrixThe solution to the problem isDetermine
- Solve the set of linear algebraic equationsusing LU decomposition. 1 11 212 3 2 2 X1 IH X2 X3 09 0 15 -3 20 -1 0 -1
- Explain the mathematical modelin the context of Eq. (1.1.1). y+c=b (1.1.4)
- Identify the operator in the following equation: ду at ах2 азу ta = -ky + x3
- Explain the mathematical modelin the context of Eq. (1.1.1) where A is a square matrix. Ay = b (1.1.8)
- Is the operator of the following equation linear? d³y dx3 + x²y = sin(x)
- What are the operator, unknown(s), forcing function and input(s) for dy az² dt +y=0
- What are the operator, unknown(s), forcing function and input(s) for T² T = 5x
- Determine the concentration profile for a batch reactor with the consumption rate r(y) = k1y.
- What are the operator, unknown(s), forcing function and input(s) forwhere z is a constant. af at a²f əx² - 6f+z = 0
- Solve Eqs. (1.2.25)–(1.2.27) for the kinetic rate constants k1 = k−1 = 2 and k2 = 3 and the initial conditions cA(0) = 1, cB(0) = 0, and cC(0) = 0. dcA dt -KICA +K-1CB (1.2.25)
- Determine the concentration profile for a batch reactor with the consumption rate r(y) = k2y2.
- Solve the following ordinary differential equation y=3xy, y(3) = 1
- Solve Eq. (1.2.60) subject to boundary conditions c(0) = 0 and c(L) = c0. d²c 0 = D - kc dx² (1.2.60)