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# Numerical Analysis

### Simply supported beam: Maximum bending moment

The bending moment M at position x m from the end of a simply supported beam of length L m carrying a uniformly distributed load of w Kn m-1 is given by M = w/2 L (L-x) - w/2 (L - x)^2 Show that the maximum bending moment occurs at the mid point of the beam, and determine its value in terms of w and L.

### Heat Equation Problem

Find the solution u(x,t) of the heat equation: ut = 1/2 uxx (a) with initial data u(x,0) = x (b) with initial data u(x,0) = x^2 (c) with initial data u(x,0) = sinx (d) with initial data u(x,0) = 0 x < 0 and u(x,0) = 1 or x >/= 0 I know the solution of the heat equation with given initial data is unique. So if you happe

### The 1D heat equation

Please see the attached file for the fully formatted problems. Solve the heat equation u_t = ku_xx for 0<x<L With boundary conditions u(0,t)=u(L,t)=0 Solve for the initial value conditions: a. u(x,0) = sin(5*Pi*x/L) b. u(x,0) = x c. For part b, plot the solution at t=0, 0.1, 1

Please see the attached file for the fully formatted problems. find a function f(x,y) such that the Grad(f) = (2xy+y^3+1, x^2+3xy^2) Explain why you cannot find f(x,y) such that Grad(f) = ( x^2+3xy^2, 2xy+y^3+1)

### Sturm-Liouville: Expansion in Eigenfunctions

Hello, I need a detailed solution of the attached problem. Consider the eigenvalue problem y" + lambda^2 * y = 0 With the boundary conditions: y(0) = y'(1)=0 1. Determine the eigenfunctions and eigenvalues. 2. show that the eigenfunctions are orthogonal. 3. Show how to expand a smooth function in terms of the

### Explain the following in your own words: (a + b)2 ¹ a2 + b2. Give a numerical example to illustrate this.

Explain the following in your own words: (a + b)2 ¹ a2 + b2. Give a numerical example to illustrate this.

### Examples of the Method of Characteristics for Solving PDEs

1. Use the method of characteristics to solve the advection equation du/dt=-kdu/dx-ru subject to the initial condition u(x,0)=f(x). 2. Use the method of characteristics to solve du/dt+te^(-t^2))du/dx=usin(t) subject to the initial condition u(x,0)=e^(-x^2)) (See attachment for the above questions formatt

### Cancellation Round-off Error : Numerical Analysis : Confirmation

I have numerically solved the following quadratic equation: 1.002x2 - 11.01x + 0.01265 = 0. IS IT POSSIBLE THAT IN THIS INSTANCE EQUATION (2) EQUALS (2A) BELOW: If b2 - 4ac >0, the quadratic equation ax2 + bx +c = zero has two real solutions x1, x2 given by the typical: (1) x1 = (-b + sqrt(b^2-4ac))/ (2a)

### Numerical Analysis : Orthogonal Polynomials and Least Squares Approximations

Following questions need to be answered: 2, 4, 6, 7, 8, 11, & 12. (page 502) See attached file for full problem description.

### Actual, Absolute and Maximum Relative Errors in Data

A touring company has the following box office takings for the recent month. Venue Clarion West Riding West End Burlington Huntingdon Erie Adelphia Wilson Reading How do I estimate the abosolute and relative maximum and average errors in the total of the above rounded data? How do I determine the actual error i

### Numerical Analysis - Fixed Point iteration method.

Fixed Point iteration method. Use a fixed-point iteration method to find an approximation to that is accurate within 10-4 See attached file for full problem description.

### Working with half life of a radioactive material

Polonium-210 has a half life of 140 days If a sample has a mass of 200 mg find a function describing the mass that remains after t days When will the mass be reduced to 10 mg?

Adam collects stamps. He has 18 bird stamps,9 flower stamps and 12 butterfly stamps. For a school project, he will display an equal number of each kind of stamp on a small poster boards. What is the greatest number of poster boards Adam can make if he uses all of the stamps?

### Numerical Analysis Problem/Secan Method

(See attached file for full problem description with proper symbols) --- A) Consider a variation of Newton's method in which only one derivative is needed; that is . Find C and s such that . B) Find the conditions on to ensure that the iteration will converge linearly ( ) to a zero of f if started near the zero.

### Numerical Stability

(See attached file for full problem description with equations) --- There is a function f of the form for which and . Determine and , and assess the sensitivity of these parameters to slight changes in the values of f at the two indicated points. ---

### Solving an IVP with Maple: Euler and Improved Euler Method

Use Maple to solve this exercise: Consider the following (IVP) logistic model p' = 10p(1-p) with p(0)=0.1 1. Solve this IVP and graph the solution over the interval [0, 10], Write down the Euler approximation, and Improved Euler approximation with step size h. 2. Compute and plot the first 100 points of the Euler method

### Solving 2 ODE's with One Unknown

Given that: dMN/dt = P¬oert - pMN + m(MS - MN) dMS/dt = m(MN - MS) - pMS¬ MN¬ (0) = MS (0) = 0 And using: M¬N + MS = [Po/(r + p)](ert - e-pt) Show that MS(t) = And show that R = MN(t) / MS(t) = Please see the attached file for the fully formatted problems.

### PDE : Riemann's Method for Solving Cauchy Problem

Hello. Thanks for help! I will use * to indicate a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. This is the probelm: Use Riemann's method to solve the Cauchy problem: u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x, on the intial curve y=2x. If this

### Wave Equation : Solve a PDE

Please see the attached file for the fully formatted problems.

### Numerical Methods, 400 Undergraduate level.

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. If there is anything unclear in the question, let me know. Thank you. (Complete problem in attachment)

### Numerical Methods: Fixed-Point Iteration

Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Let g(x) = 0.5x + 1.5 and p0 = 4, and consider fixed-point iteration. a) Show that the fixed point is P=3 b) Show that |P - Pn| = |P - Pn-1|/2 for n = 1, 2, 3... c) Show that |P - Pn| = |P - P0|/2^2 for n = 1, 2, 3...

### Three=digit rounding arithmetic

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. 7. Use three-digit rounding arithmetic to compute the following sums... Please see attached.

### Existence and uniqueness theorem

I am looking for the solution of this problem. It involves a little bit of theory in the second part. In the solution give a detailed solution showing all assumptions and theorems.

### Partial Differential for Diffusion Equation

Suppose that u(x,t) satisfies the diffusion equation... for 0<x<L and t>0, and the Robin boundary conditions... where k, L, a0 and aL are all positive constants. Show that... is a decreasing function of t. Please see attached for full question.

### Solving the Initial Value Problem

Please solve the initial value problem. See attached file.

### Numerical Methods - Spline

Consider the natural cubic spline function s(x) interpolation the following data... Please see attached for full question.

### Heat Equation, Boundary & Steady State Conditions, Initial Value

Please help with the following problems involving numerical analysis. (a) Find all the separated solutions of the attached heat equation (satisfying the attached boundary condition) (b) Use these separated solutions to write a series solution for the initial value problem posed by the attached pde and the attached boundar

### ELEMENTARY NUMERICAL ANALYSIS by ATKINSON HAN

Please see the attached file for full problem description.

### Using Malthus Method

Suppose that a culture of bacteria has initial population of n=100. If the population doubles every three days, determine the number of bacteria present after 30 days. How much time is required for the population to reach 4250 in number

### Dipoles, Quadrupoles and Laplacian: Limiting Potential

Consider a dipole of strength D, oriented along the x-axis and located at the point x =E/2 , y = 0, z = 0, and a dipole of strength -D, oriented along the x-axis at the point x = - , y = 0, z = 0. Write the expression for the potential for these two dipoles. Take the limit as , and show that the limiting potential is given by