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Module Detailed Contents Hrs., Module: Laplace Transform 07 Hrs., 1.1 Definition of Laplace transform, Condition of Existence of. Laplace transform,, 1.2 Laplace Transform (L) of Standard Functions like e“', sin(at), cos(at),, sinh(at), cosh(at) and t" , where n= 0., 01 | |:3 Properties of Laplace Transform: Linearity, First Shifting theorem, Second|, Shifting Theorem, change of scale Property, multiplication by 1, Division by |, Laplace Transform of derivatives and integrals (Properties without proof)., 1.4 Evaluation of integrals by using Laplace Transformation., learning topics: Heaviside’s Unit Step function, Laplace Transform. of Periodic, functions, Dirac Delta Function., Module: Inverse Laplace Transform 06 Hrs., 2.1 Inverse Laplace Transform, Linearity property, use of standard formulae to, find inverse Laplace Transform, finding Inverse Laplace transform using, derivative, 02 | 2.2 Partial fractions method & first shift property to find inverse Laplace transform], 2.3 Inverse Laplace transform using Convolution theorem (without proof), Self-learning Topics: Applications to solve initial and boundary value problems, involving ordinary differential equations., Module: Fourier Series: 07Hrs., 3.1 Dirichlet’s conditions, Definition of Fourier series and Parseval’s Identity, (without proof), 3.2 Fourier series of periodic function with period 2x and 2),, 03 | 3.3 Fourier series of even and odd functions, 3.4 Half range Sine and Cosine Series., Self-learning Topics: Complex form of Fourier Series, orthogonal and orthonorm:, set of functions, Fourier Transform., jodule: Complex Variables: 07Hrs., .1 Function f(z) of complex variable, limit, continuity and differentiability of Az),, 04 Analytic function, necessary and sufficient conditions for Az) to be analytic (without proof),, 2 Cauchy-Riemann equations in cartesian coordinates (without proof), 3 Milne-Thomson method to determine analytic function f(z) when real part (u) or Imaginary|, (v) or its combination (uty or u-v) is given., , , , , , , , Page 9 of 77
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4.4 Harmonic function, Harmonic conjugate and orthogonal trajectories, , Self-learning Topics: Conformal mapping, linear, bilinear mapping, cross ratio, fixed points and|, standard transformations, , Module: Matrices: 06 Hrs., , , , , , 5.1 Characteristic equation, Eigen values and Eigen vectors, Properties of Eigen, values and Eigen vectors. (No theorems/ proof), , 5.2 Cayley-Hamilton theorem (without proof): Application to find the inverse, of the given square matrix and to determine the given higher degree, , 05 polynomial matrix., , 5.3 Functions of square matrix, , 5.4 Similarity of matrices, Diagonalization of matrices, , Selflearning Topics: Verification of Cayley» Hamilton theorem, Minimal, polynomial and Derogatory matrix & Quadratic Forms (Congruent transformation &], {Orthogonal Reduction), , Module: Numerical methods for PDE. 06 Hrs., , 6.1 Introduction of Partial Differential equations, method of separation of, variables, Vibrations of string, Analytical method for one dimensional heat and, 06 wave equations. (only problems), , 6.2 Crank Nicholson method, , 6.3 Bender Schmidt method, , Self-learning Topics: Analytical methods of solving two and three dimensional, , problems., Total 39, , , , , , , , , , , , , , , , 9°, , Term Work:, , General Instructions:, , 1 Batch wise tutorials are to be conducted. The number of student’sperbatch should be as per University, pattern for practicals., , 2 Students must be encouraged to write at least 6 class tutorials on entire syllabus., , 3A group of 4-6 students should be assigned a self-learning topic. Students should prepare a, presentation/problem solving of 10-15 minutes. This should be considered as mini project in, Engineering Mathematics. This project should be graded for 10 marks depending on the, performance of the students., , Page 10 of 77
