1. a. Derive the 2nd order finite difference approximation of f'(x) (2) Taylor series expansion. using...

 

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1. a. Derive the 2nd order finite difference approximation of f'(x) (2) Taylor series expansion. using [2 marks] b. Discuss the order truncation error of finite difference approximations. Include how the order of truncation error effect the error when halving the step size h. [4 marks] c. The steady-state heat conduction governing equation in radial cylindrical coordinates can be expressed as: dT 1 dT + 0 dr r dr Where T is the temperature and r the radial coordinate. A thick-walled pipe of outer radius a, and inner radius a/2 is exposed to a constant temperature of Ta/2 = 0 on the inner surface and Ta = 100 on the outer surface. By using the non-dimensional parameter x = r/a and step size Ax = 0.1, derive the system of equations in Ax=b form, where A is a 4 x 4 matrix and using central finite difference approximations. [10 marks] d. With respect to your answer in Part C, how would the system change if a heat flux (Von Neumann) type boundary condition is applied to the inner and outer surfaces. What would the size of the coefficient matrix A be? [4 marks] 1. a. Derive the 2nd order finite difference approximation of f'(x) (2) Taylor series expansion. using [2 marks] b. Discuss the order truncation error of finite difference approximations. Include how the order of truncation error effect the error when halving the step size h. [4 marks] c. The steady-state heat conduction governing equation in radial cylindrical coordinates can be expressed as: dT 1 dT + 0 dr r dr Where T is the temperature and r the radial coordinate. A thick-walled pipe of outer radius a, and inner radius a/2 is exposed to a constant temperature of Ta/2 = 0 on the inner surface and Ta = 100 on the outer surface. By using the non-dimensional parameter x = r/a and step size Ax = 0.1, derive the system of equations in Ax=b form, where A is a 4 x 4 matrix and using central finite difference approximations. [10 marks] d. With respect to your answer in Part C, how would the system change if a heat flux (Von Neumann) type boundary condition is applied to the inner and outer surfaces. What would the size of the coefficient matrix A be? [4 marks]