Course Notes
ASSIGNMENT - F SLOT
1. Solve the ordinary linear differential equation d2T/dx2 +1000 x2 = 0 , 0 ≤x≤1 subject to boundary conditions T(0)=0 , T(1) = 0 .Obtain the approximate solution using the following methods. Consider two parameter solution (i) Collocation Method , collocation points x=1/3 , & x= 2/3 . (ii) Sub domain Method , Sub domain : 0-1/2 & 1/2 - 1 . Check the approximate values with exact values @ x=1/4 & 3/4 2. Consider the differential equation d2T/dx2 + 100 = 0 , 0≤x≤10 subjected to the boundary conditions T(0) = T(10) =0 Assume two parameter solution using trigonometric functions and using the following methods (i) Least square method (ii) Galerkin method Check the approximate values with exact values at x= 1/3 and 2/3 . 3. 4. Apply Galerkin Method to the boundary value problem Y” +Y+x =0 (0≤x≤1) ; y(0) = y(1) =0 to find approximate solution . Consider two parameter solution compare the approximate values to exact values at x= 0.5 Consider the differential equation -d2u/dx2 +u-cos(πx) =0 Subjected to the following two sets of boundary conditions u(0) =0 ,(du/dx )x=1 = 0 (du/dx)x=0 =0 , (du/dx)x=1 =0 Determine a three parameter solution with trigonometric function using (a) Galerkin method (b) Least square method Approximate the following derivatives in the finite difference form d2y/dx2 ,d3y/dx3 , d4y/dx4 Determine values of y at the pivotal points of the interval (0,1) at y satisfies the boundary value problem Y’’’ + 81 y – 81x2 = 0 ;y(0) =y(1) = y”(0) =y”(1) = 0 Take 3 meshes . Using the finite difference method obtain the solution for the following differential equation d2y/dx2 +y = x subject to the boundary conditions y(0)=0 ; (dy/dx)x=1 =2 .Take 3 meshes . Obtain the exact solution and compare approximate value to exact value at x= 0.5 . Write a programme in MATLAB to obtain solution for the following differential equation in a square region .Divide the region into ‘n’ no of meshes . ˅2 Y = 0 The boundary values are of the following Left side -
1. Solve the ordinary linear differential equation d2T/dx2 +1000 x2 = 0 , 0 ≤x≤1 subject to boundary conditions T(0)=0 , T(1) = 0 .Obtain the approximate solution using the following methods. Consider two parameter solution (i) Collocation Method , collocation points x=1/3 , & x= 2/3 . (ii) Sub domain Method , Sub domain : 0-1/2 & 1/2 - 1 . Check the approximate values with exact values @ x=1/4 & 3/4 2. Consider the differential equation d2T/dx2 + 100 = 0 , 0≤x≤10 subjected to the boundary conditions T(0) = T(10) =0 Assume two parameter solution using trigonometric functions and using the following methods (i) Least square method (ii) Galerkin method Check the approximate values with exact values at x= 1/3 and 2/3 . 3. 4. Apply Galerkin Method to the boundary value problem Y” +Y+x =0 (0≤x≤1) ; y(0) = y(1) =0 to find approximate solution . Consider two parameter solution compare the approximate values to exact values at x= 0.5 Consider the differential equation -d2u/dx2 +u-cos(πx) =0 Subjected to the following two sets of boundary conditions u(0) =0 ,(du/dx )x=1 = 0 (du/dx)x=0 =0 , (du/dx)x=1 =0 Determine a three parameter solution with trigonometric function using (a) Galerkin method (b) Least square method Approximate the following derivatives in the finite difference form d2y/dx2 ,d3y/dx3 , d4y/dx4 Determine values of y at the pivotal points of the interval (0,1) at y satisfies the boundary value problem Y’’’ + 81 y – 81x2 = 0 ;y(0) =y(1) = y”(0) =y”(1) = 0 Take 3 meshes . Using the finite difference method obtain the solution for the following differential equation d2y/dx2 +y = x subject to the boundary conditions y(0)=0 ; (dy/dx)x=1 =2 .Take 3 meshes . Obtain the exact solution and compare approximate value to exact value at x= 0.5 . Write a programme in MATLAB to obtain solution for the following differential equation in a square region .Divide the region into ‘n’ no of meshes . ˅2 Y = 0 The boundary values are of the following Left side -