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(a) If V = x3 + axy², where a is a constant, show that ле + y av ду = 3V ax Find the value of a if V is to satisfy the equation a2v a2v Əx² ду?

(a) If V = x3 + axy², where a is a constant, show that ле + y av ду = 3V ax Find the value of a if V is to satisfy the equation a2v a2v Əx² ду?

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(a) If V = x³ + axy², where a is a constant, show that +y = 3V əx ду Find the value of a if V is to satisfy the equation a²v__a²v = 0 ду? əx² (b) Show that wave equation is satisfied when a² = b²c² 1 a²u a²u Wave equestion: c² at² əx² u = cos at sin bx (c) Determine the first three non-zero terms of the Taylor series expansion for the given function. f (x) = e2× cos(x) about x =0 (d) The partial differential equation a²u a²u 3D 16 — х2 — 2у for 0 < x < 4, 0 < y< 2 (1.1) ax² ду? is subject to the boundary conditions u(x, 0) = 0 and u(x, 2) = 2(16 – x²) for 0 <x <4 u(0, y) = y and u(4,y) = 0 for 0 < y < 2 Using centred difference approximations with a grid size of h = 1, write the above boundary value problem in finite difference form, Sketch the finite difference grid and input the boundary conditions and label the unknown nodes. Hence show that the finite difference form of equation (1.1) can be written as: -4 1 01 ru11] -4 1|u21| = 1 1 -14 -9 -4] lu31. Using gausse ellimination method solve above system of equations for u11, u12 and u13.
Transcribed Image Text:(a) If V = x³ + axy², where a is a constant, show that +y = 3V əx ду Find the value of a if V is to satisfy the equation a²v__a²v = 0 ду? əx² (b) Show that wave equation is satisfied when a² = b²c² 1 a²u a²u Wave equestion: c² at² əx² u = cos at sin bx (c) Determine the first three non-zero terms of the Taylor series expansion for the given function. f (x) = e2× cos(x) about x =0 (d) The partial differential equation a²u a²u 3D 16 — х2 — 2у for 0 < x < 4, 0 < y< 2 (1.1) ax² ду? is subject to the boundary conditions u(x, 0) = 0 and u(x, 2) = 2(16 – x²) for 0 <x <4 u(0, y) = y and u(4,y) = 0 for 0 < y < 2 Using centred difference approximations with a grid size of h = 1, write the above boundary value problem in finite difference form, Sketch the finite difference grid and input the boundary conditions and label the unknown nodes. Hence show that the finite difference form of equation (1.1) can be written as: -4 1 01 ru11] -4 1|u21| = 1 1 -14 -9 -4] lu31. Using gausse ellimination method solve above system of equations for u11, u12 and u13.
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