Objectives: 1. Solve a boundary value problem (BVP) numerically using the finite difference method. 2. Explore different

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Objectives 1 Solve A Boundary Value Problem Bvp Numerically Using The Finite Difference Method 2 Explore Different 1 (36.88 KiB) Viewed 40 times

Objectives: 1. Solve a boundary value problem (BVP) numerically using the finite difference method. 2. Explore different finite difference approximations in this context, I want to see central differencing scheme with different accuracies). 3. Explore the implemented numerical solution with different BVP parameters (i.e., material properties and loads) and mesh sizes. 4. Plot relevant results including convergence (or lack of) of the numerical solution with decreasing mesh size. 5. Include any references you have used.
a The ODE that models heat conduction across a 1D rod just like the one depicted in problem 1 is given by: d dx (KA(X) OR +s(x) = 0 -53-6 dt dx 0<x<L The rod's temperature is fixed on the left end and loses heat through air convection on the left end. This ODE is a restatement of energy balance. Here: • T is temperature along the x-axis • k is the thermal conductivity o Typical values: 100-500 W/m/C A(x) is the area of cross-section o This can be constant, piece-wise constant, a linear or a quadratic function with the min and max areas of cross section being 0.05 mʻand 0.1 m², respectively s(x) is heat generated/withdrawn at x per unit length o This can be constant, linear, or quadratic with the min and max values being 0 and 10 W/m • L is the length of the bar o Assume this is 10 m The boundary conditions are: T(x = 0) = 300° C = kāx lx=z =-h(T(L) – Tair) (() dT dx =L Here, h is the convection coefficient (typical values for air are 2-20 W/m²/C and Tair is the air temperature (assume 20° C).