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Numerical Analysis

Rectangular Drum Problem #9

Just #9, please. Solve the rectangular drum problem u_xx + u_yy = (c^-2)u_tt 0 < x < a 0 < y < b t > 0 u(x,y,t) = 0, u(a,y,t) = 0 0 < y < b t > 0 u(x,0,t) = 0, u(x,b,t) = 0 0 < x < a t> 0 u(x,y,0) = xy and (u_t)(x,y,0) = 0. Find the solution explicitly in the case

ODEs: Sturm-Liouville Form; Weight Function; Inner Product

Write each of these ODEs in Sturm-Liouville form, and identify the weight function that would be used in defining an appropriate inner product {see attachment for ODEs} Use words to describe the solution process. Use appropriate math symbol editor like LateX. Please no stuff like <=.

Initial Value Problem (IVP); Euler's Method; Step Sizes

4. Consider the initial value problem (IVP): y'(t) = 3+t+y y(0)=1 a) Approximate y(1) using Euler's method and step sizes of 0.2. Perform these calculations by hand. What is the exact value of y(1)? b) Use the computer (e.g. ODE Architect, ODE Toolkit, or your own program) to approximate y(1) using step sizes of 0.1, 0.05,

Numerical Euler's Method for Initial Value Problem

Consider the initial value problem (IVP): y'(t) = y^2 y(0)=1 Approximate y(1) using Euler's method and step sizes of 0.25. Perform these calculations by hand (using a calculator for arithmetic is ok). What is the true value of y(1)?

Solve: A Second Order ODE

Please solve for the following: Find a particular solution to y'' + 5y' +4y = -13te^(3t) Show all work.

Carrying capacity of a biologists

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7000 . The number of fish doubled in the first year. dP/dt = rP(1- p/K) find an expression for the size of the population after t years by determining constant r.

Important information about mixture

A tank contains 1320L of pure water.A solution that contains .o1kg of sugar per liter enters a tank at the rate 3L/min The solution is mixed and drains from the tank at the same rate. Solve for function of t So far I have the equation:

Bisection and newton's method

Q Using the bisection method, find the positive root of 2x(1 + x^2)^-1 = arctan x. Using this root as x0; apply Newton's method to the function f(x) = arctan x: Interpret the results you obtain.

Minimal Surface Equation

Please see the attached file for the fully formatted problems. Show that Z(x, y) = ln(sin y/sin x) is a solution to the minimal surface equation. (1 + Z)Z1 + 2ZXZZX + (1 + Z)Z = 0, in the region 0 < x < ir, 0 < y < pi. What happens on the boundary of this region? Suppose we consider a constant multiple of Z(x, y) ? is i

Mathematics: Numerical Analysis Sample Question

PDE- SEND ANSWER AS ATTACHMENT What happens on the boundary of the region? Suppose we consider a constant multiple of Z(x, y). Is it still a solution of the PDE? See attachment for question and details

Elliptic Boundary Value Problem: Laplace and Polar Coordinates

(lap) means the Laplacian Vrr means the second derivative of V with respect to r V(theta theta) means the second derivative of V with respect to theta Solve: (lap)V(r,theta)= Vrr+(1/r)Vr+(1/r^2)V(theta theta)=0 0 < r < 1, -(pi) < theta < pi V(1,theta) = {1, -(pi/2) < theta < (pi/2) {0, elsewhere Ple

Elliptic Boundary Value Problem

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < pi U(x,0) = 0, U(x,pi) = 1, 0 < x < pi U(0,y) = 0, U(pi,y) = 1 0 < y < pi I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides fo

Solve an IVP ODE using the Method of Variation of Parameters

Please see the attached file for the fully formatted problems. Solve an IVP ODE using the method of variation of parameters Find the solution of the system X' using the method of variation of parameters 2 0 0 cos(t) X' = -1 0 -1 X + sin(t) 1 1 2 e^-t that satisfies the int