########## Semester4 PaperTitle:MathematicalPhysics-II Paper Code: PHYMAJ203-4 Credit:04(3+0+1) Lecture:45

Objectives: main objective of this paper to give concepts about Fourier series, method of solution Objectives: main objective of this paper to give concepts about Fourier series, method of solution of ordinary and partial differential equations, special integrals and errors.

########## Courseoutcomes

: successfulcompletionofthis course,studentswill beable to: : successfulcompletionofthis course,studentswill beable to:

- 1. functions in terms of Fourier series. 1. functions in terms of Fourier series.
- 2. order ordinary differential equations using Frobenius method 2. order ordinary differential equations using Frobenius method
- 3. the solutions of Legendre, Bessel, Hermite and Laguerre Differential Equations and their 3. the solutions of Legendre, Bessel, Hermite and Laguerre Differential Equations and their application in various problems related to physics.
- 4. -ordinatesystems using 4. -ordinatesystems using separation of variables method.
- 5. Functionsandexpress integralsin termsofthem. 5. Functionsandexpress integralsin termsofthem.

########## Unit I

Fourier Series Periodic functions. Orthogonality of sine and cosine functions, Dirichlet Conditions Fourier Series Periodic functions. Orthogonality of sine and cosine functions, Dirichlet Conditions Series (Statement only). Expansion of periodic functions in a series of sine and cosine functions and determination of Fourier coefficients. Complex representationof Fourier series. Expansion of functions with arbitrary period. Expansion of non- periodic functions over an interval. Even and odd functions and their Fourier expansions. Application. Summing of Infinite Series. (7 Lectures)

########## Unit II

Frobenius Method and Special Functions Singular Points of Second Order Linear Differential Frobenius Method and Special Functions Singular Points of Second Order Linear Differential Method and Special Functions Equations and their importance. Frobenius method and its applications to differential equations. Legendre, Bessel, Hermite and Laguerre Differential Equations. Properties of Legendre Polynomials: Rodrigues Formula, Generating Function, Orthogonality. Simple recurrence relations. Expansion of function in a series of Legendre Polynomials. Bessel Functions of the First Kind: Generating Function, simple recurrence relations. Zeros of Bessel Functions (J (x) and J (x)) and Orthogonality. o 1

########## (20 Lectures)

########## Unit III

SomeSpecialIntegrals: Integrals in SomeSpecialIntegrals: Integrals in terms of Gamma Functions. Error Function (Probability Integral). (4 Lectures) SystematicandRandomErrors.PropagationofErrors.NormalLawofErrors. Standard TheoryofErrors: and Probable Error. Least-squares fit. Error on the slope and intercept of a fitted line. (4 Lectures) Lectures) :Solutionstopartialdifferentialequations,usingseparationof PartialDifferentialEquations

variables: Laplace's Equation in problems of rectangular, cylindrical and spherical symmetry. Wave equation and its solution for vibrational modes of a stretched string, rectangular and circular membranes. Diffusion Equation. (10 Lectures)