BSc Nautical Sciences APPLIED MATHEMATICS III Syllabus - Mumbai University 2026

Module I: Bessel Functions, Legendre Polynominals&

• PartialDifferential Equations: Relations between Laplace equation and Bessel’sdifferential equation, Its solution by series methods, Besselfunctions of first and second kind, Recurrence relations forJ (x), Relation between Laplace equation and Legendre differential equation, Itssolution by series methods, Recurrence relations for Pn(x),Rodriguez’s formula for P(x), Partial differential equation governing Transverse Vibrationsof an elastic string, its solution using Fourier Series,Vibrations of a rectangular and circular membrane. Heatequation, steady – state configuration for heat flow andLaplace equation in two and three dimensions, Variableheat flow in one dimension.

Module II: Laplace Transforms

• Function of bounded variation (Statement only), Laplace transforms of 1, t n ,e at , sin (at), cos (at), Sin h (at), Cos h(at), erf (t), Shifting properties. Expressions (with Proofs) for : (i) L{t n f(t)} (ii) L{f(t)}/T (iii) L{ꭍ f (u)du}(iv) L{d n f(t)}/dt n Evaluation of inverse Laplace Transforms, partial fractionmethods, convolution theorem. Application to solve initial and boundary value problems involving ordinary differential equations with onedependent variable.

Module III: Complex Variables

• Functions of complex variable. Continuity (only statement)derivability of a function analytic. Regular function.Necessary conditions for f (z) to be analytic. (Statement ofsufficient conditions). Cauchy Riemann equation in polarco-ordinates. Harmonic functions, Orthogonal trajectories.Analytical and Milne – Thomson method to find f (z) from itsreal or imaginary parts. Integration of complex functions,Cauchy’s integral theorem for simply connected regions, Cauchy’s integral formula, Taylor’s and Laurent’sexpansion, Zeros, Singularities, poles, residue at isolatedsingularity and its evaluation. Residue theorem, itsapplication to evaluate real integrals.

Reference Books

This paper runs across both semesters of the year and the University prints one list of reference books for it, at the end of the second half. Her list is repeated here so that this half is not read as a paper she recommends no reading for.

• 1 A Text book for applied mathematics Wartikar P.N. &WartikarJ.N.
• 2 A Text Book of Matrices Shanti Narayan
• 3 Mathematical Statistics J.N.Kapur, H.C. Saxena
• 4 Statistics in Schaum’s Series Murray R.Spiegal
• 5 Probability& Statistics for engineers Myers
• 6 Higher Engineering Mathematics Dr. B.S. Grewal
• 7 Numerical methods for engineers S.K. Gupta
• 8 Operation Research, An introduction H.A. Taha
• 9 Operation research methods & Problems Sasieni, Yaspan, Friedman
• 10 Linear Programming G. Hadley