Course objective
The objective of this course is to understand basic methods for solving nonlinear first order ordinary differential equations and existence of solutions, along with some special types of second order ordinary differential equations of mathematical physics. Also, students will be exposed to second order partial differential equations arising in thermal physics and thermodynamics.
Unit I
Existence and Uniqueness of Solutions: Lipschitz condition, Gronwall type inequality, successive approximations, Picard's theorem, non-uniqueness of solutions, continuation and dependence on initial conditions, existence of solutions in the large.
Unit II
Solution of second order ODE with constant coefficients, power series solutions of ordinary and singular points, and special functions of Legendre's differential equations, Bessel's differential equations and their properties.
Unit III
Charpit's method, special types of first order PDE, Jacobi's method, linear second order PDE, canonical forms of second order PDE and characteristic curves, one dimensional wave equation, its origin and elementary solutions, vibration of an infinite string, vibration of a semi-finite string, vibration of a string of finite length, existence of unique solution.
Unit IV (Practical)
Laboratory work for the following problems using MATLAB / Mathematica / Maple etc.: 1) Plot the Fourier series of the following functions: f(x) = x², x ∈ [−1, 1]; f(x) = 1 for 0 < x < π, f(x) = −1 for −π < x < 0; f(x) = sin x, 0 < x < π/2. 2) Solution of wave equation ∂²u/∂t² − c² ∂²u/∂x² = 0 for the following associated conditions: (i) u(x,0) = φ(x), uₜ(x,0) = σ(x), x ∈ R, t > 0; (ii) u(x,0) = φ(x), uₜ(x,0) = σ(x), u(0,t) = 0, x ∈ (0,∞), t > 0; (iii) u(x,0) = φ(x), uₜ(x,0) = σ(x), uₓ(0,t) = 0, x ∈ (0,∞), t > 0; (iv) u(x,0) = φ(x), uₜ(x,0) = σ(x), u(0,t) = 0, u(l,t) = 0, 0 < x < l, t > 0. 3) Solution of one dimensional heat equation ∂u/∂t − k ∂²u/∂x² = 0 for the following conditions: (i) u(x,0) = φ(x), u(0,t) = a, u(l,t) = b, 0 < x < l, t > 0; (ii) u(x,0) = φ(x), x ∈ R, 0 < t < T; (iii) u(x,0) = φ(x), u(0,t) = a, x ∈ (0,∞), t ≥ 0.
Learning outcomes
- Understand first order nonlinear ordinary differential equations and existence of solutions.
- Learn the methods to find solutions of second order linear ordinary differential equations with constant coefficients and variable coefficients.
- Learn the different methods for solving first and second order partial differential equations, and be able to take more courses on wave equation, heat equation, diffusion equation, gas dynamics, nonlinear evolution equations, etc. All these courses are important in engineering and industrial applications for solving boundary value problems.
- Get idea to solve various mathematical models of ODE and PDE which may be helpful for simulation process.
Books recommended
- Deo and Raghavendra, Text Book of Ordinary Differential Equations, Tata McGraw-Hill Pub. Company Ltd., New Delhi, 2017.
- G. F. Simmons, Differential Equations, Tata McGraw-Hill, 1991. (Ch-5: 28-30, Ch-8: 44-47, Ch-6: 33-36)
- I. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, International Students Edition, 2006. (Ch-2: 10, 11, 13; Ch-3: 4-7; Ch-5: 1, 2)
- Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, 4th edition, Birkhauser, Indian reprint, 2014.
Books for reference
- J. N. Sharma and Kehar Singh, PDE for Engineers and Scientists, Narosa, New Delhi, 2009.
- T. Amarnath, An Elementary Course in Partial Differential Equations, Narosa Publications, 2003.
- Martin Braun, Differential Equations and their Applications, Springer International Student Ed., 1978.
- S. L. Ross, Differential Equations, 3rd Edition, John Wiley and Sons, India, 2014.
- C. Y. Lin, Theory and Examples of Ordinary Differential Equations, World Scientific, 2011.
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