Partial Differential equations of first order: Non linear partial differential equations of first order, Cauchy’s method of characteristics, - Compatible system of first order equations – Charpit’s method, Special types of first order equations.
Linear Partial Differential equations with constant coefficients - Equations with variable coefficients - Separation of variables.
Occurrence of Laplace equations in Physics - Elementary solution of Laplace’s equation- Boundary value problem-separation of variables-Problems with Axial Symmetry – Kelvin’s Inversion theorem – The theory of Green’s Function for Laplace’s Equation – The two dimensional Laplace equation- Green’s Function for the Two –dimensional Equation.
The wave Equation - The Occurrence of the Wave equation in physics- Elementary Solution of the One-dimensional Wave equation-Vibrating Membranes: Application of the Calculus of Variations-Three dimensional problem- General Solutions of the Wave equation-Green’s Function for the Wave equation- The Non homogeneous Wave equation.
The Diffusion Equation: The occurrence of the Diffusion Equation in Physics-The Resolution of Boundary Value Problem for the Diffusion Equation-Elementary solutions of the Diffusion Equation-Separation of variables-The use of Green’s Functions.
Reference Book:
Partial Differential Equations by C.Evans, AMS, Providence, RI, 2003.
Text Book:
Elements of Partial Differential Equations by IAN N Sneddon, McGraw Hill, 1988. Unit I: Chapter 2: Sections 2.7, 2.8, 2.9, 2.10 and 2.11. Unit-II: Chapter 3: Sections 3.4, 3.5 and 3.9. Unit-III: Chapter 4: Sections 4.1, 4.2, 4.4, 4.5, 4.6, 4.7, 4.8, 4.11 and 4.13. Unit-IV: Chapter 5: Sections 5.1, 5.2, 5.4, 5.5, 5.6, 5.7 and 5.8. Unit-V: Chapter 6: Sections 6.1, 6.2, 6.3, 6.4 and 6.6.
