A simple implicit scheme for the heat equation was proposed by Laasonen (1949). The algorithm for this scheme is u+¹-u At α u-2u+¹+u (Ax)² If we make use of the central-difference operator 8uu1-2u+u₁-1 we can rewrite Equation 4.92 in the simpler form: u-u At - ) = [/2a²4² + ut – Qlixx ² At+ α(4x)² 12 8u+1 (Ax)² This scheme has first-order accuracy with a T.E. of O[(At)², (Ax)² ] and is unconditionally stable. Upon examining Equation 4.93, it is apparent that a tridiagonal system of linear algebraic equations must be solved at each time level n +1. The modified equation for this scheme is + [10²³(At)² + 1 20²A² (Ax)² + 360° 300 (4x)* Ju .... (4.92) *** (4.93) (4.94)

using MATLAB

Suppose a aluminum rod of 20 ft and solid square cross section of 0.5 inches, isfixed at a wall on the left end. On the right end it is unsupported. The rod will bendbased on it’s own weight given that it has uniform density.a.) Find the Youngs modulus, Moment of Inertia, and Weight per unit necessary tosolve this problem. Then put them together to find the coefficient on the x4term.b.) Solve for coefficients c1 through c4 and report the equation governing the shapethis rod will take.c.) How far down from level does the end of the aluminum rod hang in inches?

DEVELOP THE FOLLOWING CASTIGLIANO THEOREM EXERCISE STEP BY STEP, PLACING THE MISSING STEPS

Pls help with detailed steps. Dont copy from anywhere. I want to understand the process please. Thanks!

Please show each and every step while solving. Thanks!