Assignments

Week 1:

Q1. Solve the following linear equations using Cramer’s rule.

                 2X1−3X2+X3=1

                 3X1+X2−X3=2

                   X1−X2−X3=1

Q2. Solve the following system of equations by Matrix inversion method.

                                3x+y+2z = 3

                                2x-3y-z = -3

                                x+2y+z = 4

Week 2:

Q1. What is the necessity of pivots elements in Gauss elimination method? What is meant by diagonally dominant matrix?

Q2. Solve the following system of equations using Gauss elimination method

                                x+2y+z =0

                                2x+2y+3z =3

                                -x-3y         =2

Q3. Solve the following system of equations using Gauss elimination method

                        x+ y+ z=9

                        2x-3y+4z=13

                        3x+4y+5z=40

Week 3:

Q1. Find the solutions of the following system of equations by L-U factorization method

                                2x−3y+10z=3

                                −x+4y+2z=20

                                5x+2y+z=−12

Q2. Find the solutions of the following system of equations by L-U factorization method

                                X1+2X2+3X3=5

                                2X1−4X2+6X3=18

                                3X1−9X2−3X3=6

Week 4:

Q1. State sufficient condition for convergence of Gauss-Seidel method.

Q2. Solve the following system of equations using Gauss Seidel method (correct to four places of decimals)

                        10x+2y+z=9

                         x+10y−z= −22

                        −2x+3y+10z=22

Week 5:

Q1. Solve the following system of equations using Gauss Seidel method (correct to two decimal places)

                          3x+y+5z=13

                         5x−2y+z= 4

                         x+6y−2z=−1

Q2. Find the inverse of a matrix A by using Gauss-Elimination method.

                     a1=2  a2=3  a3= -1

                     a3=4  a4=4  a5= -3

                     a6=2  a7= -3  a8=1

Week 6:

Q1. Find the solution of the following differential equation by Euler’s method for x=1 by taking h=0.2

                                 dy/dx= x+y, y=1 when x=0.

Calculate upto four significant places. Compare your result with exact solution.

Q2. Find y(0.10) and y(0.15) by Euler’s method, from the differential equation dy/dx=x2+y2 with y(0)=0, correct upto four decimal places, taking step length h=0.05.

Week 7:

Q1.    dy/dx+(y/x)= 1/x2, y(1)=1, Evaluate y(1.2) by modified Euler’s method (Euler-Cauchy method), correct upto 4 decimal places [h=0.1].                         

Q2. Using Taylor’s method, obtain an approximate value of y at x=0.2 for the differential equation, 

                                    dy/dx=2y+3ex, y(0)=0

Week 8:

Q1.   Use Runge-Kutta method of order two to find y(0.2) and y(0.4) given that,

                         y (dy/dx)= y2 – x, y(0)=2, taking h=0.2

Q2. Find y(1.1) using Runge-Kutta method of forth order, given that,

                       dy/dx= y2+xy, y(1)=1

Week 9:

Q1.   Compute y(0.8) by Adams-Moulton predictor-corrector method from

        dy/dx= 1+y2, y(0)=0, given y(0.2)=0.2027, y(0.4)=0.4228, y(0.6)=0.6842.

Q2. Find y(0.2) using Runge-Kutta method of fourth order, given that

                       dy/dx = 1+y2, y(0)=0.

Week 10:

Q1.   Compute y(0.4) by Milen’s predictor-corrector method of equation,

           dy/dx = xy+y2, given that y(0)=1, y(0.1)=1.1169, y(0.2)=1.2773, y(0.3)=1.5040.

Q2. Solve the equation

            d2y/dx2 + y = 0, with y(0)=0,y(1)=1, using Finite Difference method (FDM) taking h=0.25.