Numerical Math question

Consider a wire of length L=5 whose temperature is kept at 0^oC at both ends. Let u(x,t) be the temperature on the wire at timetand positionxwith an initial temperature (t=0) given asu0(x) =sin(x). Apartial differential equation that models the temperature distribution along the wire is

Consider a wire of length L 5 whose temperature is kept at 0C at both ends. Let u(x,t) be the temper ature on the wire at time t and position x with an initial temperature (t 0) given as uo(x) -sin(x). A partial differential equation that models the temperature distribution along the wire is with density , specific heat c, thermal conductivity K of the wire, and f(x, t) is a heat source. Using a wire made of a metal of your choice (making sure units are consistent) (a) Implement the computer program to approximate the temperatures along a wire as in example 6 of the textbook (pg. 44) using numerical derivatives. Experiment with different values of and M. You will see this approach gives results but stability issues arise if 1-2 Assume zero heat source, and solve the heat equation up to time t 3 (b) Show that the difference method for the heat equation is first-order in time and second order irn space. Hint: perform an error analysis, first by fixing the time step At and varying the step size Ax; then by fixing the step size and varying the time step M Use the first two terms (k = 2) in the analytical series solution given by Pk ! (c) Repeat the approximations for a heat source f(x,t) 3e-2t. Consider a wire of length L 5 whose temperature is kept at 0C at both ends. Let u(x,t) be the temper ature on the wire at time t and position x with an initial temperature (t 0) given as uo(x) -sin(x). A partial differential equation that models the temperature distribution along the wire is with density , specific heat c, thermal conductivity K of the wire, and f(x, t) is a heat source. Using a wire made of a metal of your choice (making sure units are consistent) (a) Implement the computer program to approximate the temperatures along a wire as in example 6 of the textbook (pg. 44) using numerical derivatives. Experiment with different values of and M. You will see this approach gives results but stability issues arise if 1-2 Assume zero heat source, and solve the heat equation up to time t 3 (b) Show that the difference method for the heat equation is first-order in time and second order irn space. Hint: perform an error analysis, first by fixing the time step At and varying the step size Ax; then by fixing the step size and varying the time step M Use the first two terms (k = 2) in the analytical series solution given by Pk ! (c) Repeat the approximations for a heat source f(x,t) 3e-2t