NUMERICAL METHODS 5.2. Maxima and minima SH603 5.3. Newton-Cote general quadrature formula Lecture : 3 Year : III 5.4. Trapezoidal, Simpson’s 1/3, 3/8 rule 5.5. Romberg integration Tutorial : 1 Part : I 5.6. Gaussian integration ( Gaussian – Legendre Formula 2 point and 3 point) Practical : 3
 6. Solution of ordinary differential equations (6 hours) Course objective: 6.1. Euler’s and modified Euler’s method To introduce numerical methods used for the solution of engineering problems. The 6.2. Runge Kutta methods for 1st and 2nd order ordinary differential equations course emphasizes algorithm development and programming and application to 6.3. Solution of boundary value problem by finite difference method and shooting realistic engineering problems. method.
 1. Introduction, Approximation and errors of computation (4hours) 7. Numerical solution of Partial differential Equation (8 hours) 1.1. Introduction, Importance of Numerical Methods 7.1. Classification of partial differential equation(Elliptic, parabolic, and Hyperbolic) 1.2. Approximation and Errors in computation 7.2. Solution of Laplace equation ( standard five point formula with iterative method) 1.3. Taylor’s series 7.3. Solution of Poisson equation (finite difference approximation) 1.4. Newton’s Finite differences (forward , Backward, central difference, divided 7.4. Solution of Elliptic equation by Relaxation Method difference) 7.5. Solution of one dimensional Heat equation by Schmidt method

1.5.Difference operators, shift operators, differential operators

1.6.Uses and Importance of Computer programming in Numerical Methods.

 2. Solutions of Nonlinear Equations (5 hours) Practical: Algorithm and program development in C programming language of following: 2.1. Bisection Method 1. Generate difference table. 2.2. Newton Raphson method ( two equation solution) 2. At least two from Bisection method, Newton Raphson method, Secant method 2.3. Regula-Falsi Method , Secant method 3. At least one from Gauss elimination method or Gauss Jordan method. Finding largest 2.4. Fixed point iteration method Eigen value and corresponding vector by Power method. 2.5. Rate of convergence and comparisons of these Methods 4. Lagrange interpolation. Curve fitting by Least square method. 5. Differentiation by Newton’s finite difference method. Integration using Simpson’s 3/8 3. Solution of system of linear algebraic equations (8 hours) rule 3.1. Gauss elimination method with pivoting strategies 6. Solution of 1st order differential equation using RK-4 method 3.2. Gauss-Jordan method 7. Partial differential equation (Laplace equation) 3.3. LU Factorization 8. Numerical solutions using Matlab.

3.4.Iterative methods (Jacobi method, Gauss-Seidel method)

3.5.Eigen value and Eigen vector using Power method

 References: 4. Interpolation (8 hours) 1. Dr. B.S.Grewal, “Numerical Methods in Engineering and Science “, Khanna 4.1. Newton’s Interpolation ( forward, backward) Publication. 4.2. Central difference interpolation: Stirling’s Formula, Bessel’s Formula 2. Robert J schilling, Sandra l harries , ” Applied Numerical Methods for Engineers using 4.3. Lagrange interpolation MATLAB and C.”, Thomson Brooks/cole. 4.4. Least square method of fitting linear and nonlinear curve for discrete data and 3. Richard L. Burden, J.Douglas Faires, “Numerical Analysis”, Thomson / Brooks/cole continuous function 4. John. H. Mathews, Kurtis Fink ,”Numerical Methods Using MATLAB” ,Prentice Hall 4.5. Spline Interpolation (Cubic Spline) 5. publication with MATLAB” , JAAN KIUSALAAS , “Numerical Methods in Engineering 5. Numerical Differentiation and Integration (6 hours) Cambridge Publication

5.1.Numerical Differentiation formulae

Evaluation scheme:

The questions will cover all the chapters of the syllabus. The evaluation scheme will be as indicated in the table below

 Unit Chapter Topics Marks 1 1 & 2 all 16 2 3 all 16 3 4 all 16 4 5 all 16 6 6.1, 6.2 5 6 6.3 16 7 all Total 80
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