v) Without calculation, sketch how the approximate solution would look if h = 1/6. Clearly show where the exact and approximate solutions will match for h = 1/6. [Total marks: 20] [2 marks] Semester 2 2023-2024 KL5002: Further Mathematics Page 8 of 8/nii) The boundary value problem is equivalent to a matrix equation Ay = b where A is a 3 x 3 matrix and b is a column vector. 10° (6390-9 ? ? Y2 = ? Using the following formulae, Aij = ['v/v/dx b; = √10 V; (x) dx determine the four missing matrix Aij and vector b; coefficients (bold ?). [4 marks] iii) Determine the approximate solution U matrix equation Ay = b (i.e. find y₁, y2 and y3). = y1V1 + y2V2 + y3V3 by solving the [5 marks] iv) The plot below shows the exact solution (pink curve). Sketch the approximate solution U(x). Clearly show where the exact and approximate solutions match. 5 u(x)=-5x(x-2) u(x) 0 h 2h X [1 mark] Semester 2 2023-2024 KL5002: Further Mathematics Page 7 of 8/nQUESTION 5 The transfer of energy through a new insulation material can be described using -u" = 10 with u(0) = u'(1) = 0. a) Show that the exact solution of the system is given by u(x) = -5x(x-2) Show all steps clearly. [5 marks] b) The finite element method (FEM), can be used to build an approximate solu- tion. The approximate solution is split into three intervals with h three functions V = y₁V₁ + y2V2 + y3V3 as shown, = using 1 V₁ V2 V3 2h 3h 0 1/3 2/3 i) Using the above figure, determine the three missing V components (bold ?): ? if 0x<h -x- 1 if h≤x≤2h V₁(x) = V2(x) = = -x+2 if h≤ x ≤ 2h |? if 2hx3h V³ (x) = { ½ x − 2 -2 if ? [3 marks] Semester 2 2023-2024 KL5002: Further Mathematics Page 6 of 8