Question

Problem 1 Consider finite difference method to approximate the solution to BVP on [O, 1] y(O) 1 With n-3. Set up the system of equations for wi, and solve the system. (You can use technology to solve the system. In this case, there is no need to show the work of solving the system.) Problem 2 Consider finite difference method to approximate the solution to BVP on [1,2] y" 18y2 y(1) 3 y(2) 3 12 With n-3. Set up the system of nonlinear equations for w and find its Jacobian.

Answer 1

y = 18y^2 y (1) = 1/3 y (2) = 1/12 y = 18y^2 partial differential^2 y/partial differential x^2 = 18y^2 The Auxiliary Equation is m^2 = 36 m = plusminus 6 m_1 = 6 & m_2 = -6 sol^n is y = Ge^m_1x + c_2 e^m_2 x y (1) = c_1 e^6 + c_2 e^-6 = 1/3 - (1) y (2) = c_1 e^12 + c_2 e^-12 = 1/12 - (2) multiply eq^n (1) by e^6 and subtracting it from eq^n (2) we get, c_1e^12 + c_2 = e^6/3 c_1e^12 + c_2 e^12 = 1/12 e^6/3 - 1/2 implies c_2 (1 - e^12) c_2 = 4e^6 - 1/12 (1 - e^12) value of c = 2.71 implies c_2 = 4 (2.71) - 1/12 (1 - (2.71)^-12 = 0.819 almostequalto 0.82 \r\n we know that c_1e^6 + c_2e^-6 = 1/3 c_1 e^6 + 09.82 e^-6 = 1/3 c_1 e^6 = 1/3 - 0.82e^-6 c_1 = 0.0008 Y = 0.000 e^6x + 0.82 theta^-6x