An experiment is performed to determine the percentage clongation, y of electrical conducting material as a function of temperature, T. The resulting data is listed in Table Q1(c) Table Q1(c): Percentage elongation, y of temperature, T T (°C) 200 250 300 375 425 475 y (% elongation) 7.5 8.6 10 11.3 12.7 8.7 Estimate the percent elongation for a temperature of 215°C by using a suitable method. (9 marks) FORMULAE Numerical Integration 3 Simpson 5 Rule : S; f(x)dx =${fo+ fm + 4 Emai fi +23°3 Simpson Rule : Sf(x)dx = h [Cf. +f1) + 3(51 + f2 +f4 + f5 + … + fn-2 + fn-1) +263 +f6+ … +fn-3] 2-point Gauss Quadrature: Sºg(x)dx = [g() +9 3-point Gauss Quadrature: S; g(x)dx = [10(-5) =[4(-5)+590 + g(1) Eigen Value Power Method : v(k+1) Av(k), k = 0,1,2…. Shifted Power Method :v(k+1) -Ashiftedv(K), k = 0,1,2…. mk+1 MR+1 Ordinary Differential Equation Fourth-order Runge-Kutta Method : Yi+1 = y; +(k + 2k2 + 2k3 + ka) where ky = hf (x,y:) kz = hf (x: + 8 +”) kz = hf (x + y + 3) kq = hf(xi + h, yi + kg) Partial Differential Equation Heat Equation: Finite Difference Method , a²u) Uj;j+1 – uij = c2Ui-1,1 – 2u; j + Ui+1.j ??2 k h2 1.j Poisson Equation: Finite Difference Method ( (a²u Ui+1.) – Zuij + Ui-1.1, Ujj+1 – 2ui j + Ui,j-1 = 0 ??2 ay2 = fij h2 k2 Wave Equation: Finite Difference Method au a2u Ui,j=1 – 2u1, +U1,1+1 = c2 Uj_1.j — Zuigj + Ui+1.) at2 ??2) k2 h2 = c2 ij FORMULAE Nonlinear equations Lagrange Interpolating: L = (x-x1)(x-X;) (X-Xn). f(x) = ?-14(x)f(x) (x2-x1)(x-2)(x2-x)’ Newton-Raphson Method : Xi+1 = x; f(x) i = 0,1,2 …. 1′(x) System of linear equations Gauss-Seidel Iteration: bi-2-41; *** dit (k+1) (k) =1420118_, Vi = 1,2,3,…,n. Interpolation Natural Cubic Spline: hk = Xk+1 -XK fk+1-1 , k = 0,1,2,3,…, n-1 hk dx = be = 6(dk+1 – dx), k = 0,1,2,3,… n-2, When; mo = 0,mn = 0, hxMx + 2(hk +hx+1)*x+1+hx+1Mx+2 = baik = 0,1,2,3,…,n – 2 Sx(x) = * (*x+1 – x)2 + “(x xx)? + Ce mos nx) (*x+1 – x) +(met hx) (x – x) k = 0,1,2,3,…n-1 Numerical Differentiation 2-point forward difference: f'(x) f(x+h)-f(x) 2-point backward difference: f'(x) = f(x)=f(x-H) 3-point central difference: f'(x) > f(x+h)-f(x-1) 3-point forward difference: f'(x) = -f(x+2)+45(x+h)-3f(x) 3-point backward difference: f'(x) = 3(x)=45(x-)+(x-2) 5-point difference formula: f'(x) =f(x+2n)+8f(x+h)-8f(x-1)+f(x-2h) 3-point central difference: f'(x) = f(x+h)-2f(x)+F(x-h) 5-point difference formula: f”(x) = -f(x+2)+16/(x+h)-30/(x)+16/(x-)-f(x-2) ? h 2h 2h 12h h2 12h2

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