MA8491 NUMERICAL METHODS Notes

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UNIT I SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS

Solution of algebraic and transcendental equations
Fixed point iteration method
Newton Raphson method
Solution of linear system of equations
Gauss elimination method
Pivoting
Gauss Jordan method
Iterative methods of Gauss Jacobi and Gauss Seidel
Eigenvalues of a matrix by Power method and Jacobi’s method for symmetric matrices.

UNIT II INTERPOLATION AND APPROXIMATION

Interpolation with unequal intervals
Lagrange’s interpolation
Newton’s divided difference interpolation
Cubic Splines
Difference operators and relations
Interpolation with equal intervals
Newton’s forward and backward difference formulae.

UNIT III NUMERICAL DIFFERENTIATION AND INTEGRATION

Approximation of derivatives using interpolation polynomials
Numerical integration using Trapezoidal,
Simpson’s 1/3 rule
Romberg’s Method
Two point and three point Gaussian quadrature formulae
Evaluation of double integrals by Trapezoidal and Simpson’s 1/3 rules.

UNIT IV INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

Single step methods
Taylor’s series method
Euler’s method
Modified Euler’s method
Fourth order Runge
Kutta method for solving first order equations
Multi step methods
Milne’s and Adams
Bash forth predictor corrector methods for solving first order equations.

UNIT V BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

Finite difference methods for solving second order two
Point linear boundary value problems
Finite difference techniques for the solution of two dimensional Laplace’s and Poisson’s equations on rectangular domain
One dimensional heat flow equation by explicit and implicit (Crank Nicholson) methods
One dimensional wave equation by explicit method.

OUTCOMES

Upon successful completion of the course, students should be able to:

  • Understand the basic concepts and techniques of solving algebraic and transcendental
    equations.
  • Appreciate the numerical techniques of interpolation and error approximations in various
    intervals in real life situations.
  • Apply the numerical techniques of differentiation and integration for engineering problems.
  • Understand the knowledge of various techniques and methods for solving first and second order ordinary differential equations.
  • Solve the partial and ordinary differential equations with initial and boundary conditions by using certain techniques with engineering applications.

TEXTBOOKS

  1. Burden, R.L and Faires, J.D, “Numerical Analysis”, 9th Edition, Cengage Learning, 2016.
  2. Grewal, B.S., and Grewal, J.S., “Numerical Methods in Engineering and Science”, Khanna Publishers, 10th Edition, New Delhi, 2015.

REFERENCES

  1. Brian Bradie, “A Friendly Introduction to Numerical Analysis”, Pearson Education, Asia,
    New Delhi, 2007.
  2. Gerald. C. F. and Wheatley. P. O., “Applied Numerical Analysis”, Pearson Education, Asia, 6th Edition, New Delhi, 2006.
  3. Mathews, J.H. “Numerical Methods for Mathematics, Science and Engineering”, 2nd
    Edition, Prentice Hall, 1992.
  4. Sankara Rao. K., “Numerical Methods for Scientists and Engineers”, Prentice Hall of India Pvt. Ltd, 3rd Edition, New Delhi, 2007.
  5. Sastry, S.S, “Introductory Methods of Numerical Analysis”, PHI Learning Pvt. Ltd, 5th
    Edition, 2015.