# Mathematics - Numerical methods of Ordinary and Partial Differential Equations

Indian Institute of Technology Kharagpur

Ordinary Differential Equations: Initial Value Problems (IVP) and existence theorem. Truncation error, deriving finite difference equations. Single step methods for I order IVP- Taylor series method, Euler method, Picard’s method of successive approximation, Runge Kutta Methods. Stability of single step methods.

Multi step methods for I order IVP - Predictor-Corrector method, Euler PC method, Milne and Adams Moulton PC method. System of first order ODE, higher order IVPs. Stability of multi step methods, root condition. Linear Boundary Value Problems (BVP), finite difference methods, shooting methods, stability, error and convergence analysis. Non linear BVP, higher order BVP.

Partial Differential Equations: Classification of PDEs, Finite difference approximations to partial derivatives. Solution of one dimensional heat conduction equation by Explicit and Implicit schemes (Schmidt and Crank Nicolson methods ), stability and convergence criteria.

Laplace equation using standard five point formula and diagonal five point formula, Iterative methods for solving the linear systems. Hyperbolic equation, explicit / implicit schemes, method of characteristics. Solution of wave equation. Solution of I order Hyperbolic equation. Von Neumann stability.

• ##### Mod-01 Lec-01 Motivation with few Examples
Dr. G.P. Raja Sekhar
• ##### Mod-02 Lec-02 Single - Step Methods for IVPs
Dr. G.P. Raja Sekhar
• ##### Mod-03 Lec-03 Analysis of Single Step Methods
Dr. G.P. Raja Sekhar
• ##### Mod-04 Lec-04 Runge - Kutta Methods for IVPs
Dr. G.P. Raja Sekhar
• ##### Mod-05 Lec-05 Higher Order Methods/Equations
Dr. G.P. Raja Sekhar
• ##### Mod-06 Lec-06 Error - Stability - Convergence of Single Step Methods
Dr. G.P. Raja Sekhar
• ##### Mod-07 Lec-07 Tutorial - I
Dr. G.P. Raja Sekhar
• ##### Mod-08 Lec-08 Tutorial - II
Dr. G.P. Raja Sekhar
• ##### Mod-09 Lec-09 Multi-Step Methods (Explicit)
Dr. G.P. Raja Sekhar
• ##### Mod-10 Lec-10 Multi-Step Methods (Implicit)
Dr. G.P. Raja Sekhar
• ##### Mod-11 Lec-11 Convergence and Stability of multi step methods
Dr. G.P. Raja Sekhar
• ##### Mod-12 Lec-12 General methods for absolute stability
Dr. G.P. Raja Sekhar
• ##### Mod-13 Lec-13 Stability Analysis of Multi Step Method
Dr. G.P. Raja Sekhar
• ##### Mod-14 Lec-14 Predictor - Corrector Methods
Dr. G.P. Raja Sekhar
• ##### Mod-15 Lec 15 Some Comments on Multi - Step Methods
Dr. G.P. Raja Sekhar
• ##### Mod-16 Lec-16 Finite Difference Methods - Linear BVPs
Dr. G.P. Raja Sekhar
• ##### Mod-17 Lec 17 Linear/Non - Linear Second Order BVPs
Dr. G.P. Raja Sekhar
• ##### Mod-18 Lec-18 BVPS - Derivative Boundary Conditions
Dr. G.P. Raja Sekhar
• ##### Mod-19 Lec-19 Higher Order BVPs
Dr. G.P. Raja Sekhar
• ##### Mod-20 Lec-20 Shooting Method BVPs
Dr. G.P. Raja Sekhar
• ##### Mod-21 Lec-21 Tutorial - III
Dr. G.P. Raja Sekhar
• ##### Mod-22 Lec-22 Introduction to First Order PDE
Dr. G.P. Raja Sekhar
• ##### Mod-23 Lec-23 Introduction to Second Order PDE
Dr. G.P. Raja Sekhar
• ##### Mod-24 Lec-24 Finite Difference Approximations to Parabolic PDEs
Dr. G.P. Raja Sekhar
• ##### Mod-25 Lec-25 Implicit Methods for Parabolic PDEs
Dr. G.P. Raja Sekhar
• ##### Mod-26 Lec-26 Consistency, Stability and Convergence
Dr. G.P. Raja Sekhar
• ##### Mod-27 Lec-27 Other Numerical Methods for Parabolic PDEs
Dr. G.P. Raja Sekhar
• ##### Mod-28 Lec-28 Tutorial - IV
Dr. G.P. Raja Sekhar
• ##### Mod-29 Lec-29 Matrix Stability Analysis of Finite Difference Scheme
Dr. G.P. Raja Sekhar
• ##### Mod-30 Lec-30 Fourier Series Stability Analysis of Finite Difference Scheme
Dr. G.P. Raja Sekhar
• ##### Mod-31 Lec-31 Finite Difference Approximations to Elliptic PDEs- I
Dr. G.P. Raja Sekhar
• ##### Mod-32 Lec-32 Finite Difference Approximations to Elliptic PDEs - II
Dr. G.P. Raja Sekhar
• ##### Mod-33 Lec-33 Finite Difference Approximations to Elliptic PDEs - III
Dr. G.P. Raja Sekhar
• ##### Mod-34 Lec-34 Finite Difference Approximations to Elliptic PDEs - IV
Dr. G.P. Raja Sekhar
• ##### Mod-35 Lec-35 Finite Difference Approximations to Hyperbolic PDEs - I
Dr. G.P. Raja Sekhar
• ##### Mod-36 Lec-36 Finite Difference Approximations to Hyperbolic PDEs - II
Dr. G.P. Raja Sekhar
• ##### Mod-37 Lec-37 Method of characteristics for Hyperbolic PDEs - I
Dr. G.P. Raja Sekhar
• ##### Mod-38 Lec-38 Method of characterisitcs of Hyperbolic PDEs - II
Dr. G.P. Raja Sekhar
• ##### Mod-39 Lec-39 Finite Difference Approximations to 1st order Hyperbolic PDEs
Dr. G.P. Raja Sekhar
• ##### Mod-40 Lec-40 Summary, Appendices, Remarks
Dr. G.P. Raja Sekhar