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GE-2313 - Numerical methods for partial differential equation Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). Unit-I contains finite difference method to solve 2" order and 4" order ordinary differential equation with three types of boundary condition. Error analysis, stability analysis and convergence analysis of finite difference methods are discussed. In many fields of science and engineering, to determine the harmonic motion, damped and forced variation, current from electric circuit, 2"! order and 4" ordinary differential equation is required to solve. The analytical solution of most of the ordinary differential equations with complicated boundary condition that occur in engineering problems is not easy. Therefore, numerical technique finite difference method (FDM) is very popular and important for solving the boundary value problems. In Unit-IJ, Il and IV , Finite difference methods are used to sove elliptic, pabalolic and hyperbolic type of partial differential equations. Partial differential equations (PDEs) arise in every field of science and engineering like hydrodynamics, elasticity, quantum mechanics and electromagnetic theory. A physical problem in applied mathematics or science and engineering can be formulated in terms of PDE, so the solution of these PDEs is of great interest in understanding various physical phenomena. Many PDEs cannot be solved by analytical methods, we go in for sufficiently approximate solution by simple numerical methods, and the method of finite differences is commonly used. Unit- II contains classification of partial differential equations, Elliptic Equations like Laplace equation, Poisson equation, iterative schemes, Dirichlet’s problem, Neumann problem, mixed boundary value problem, ADI methods. Unit-III contains Schmidt’s two level, multilevel explicit methods, Crank-Nicolson’s two level, multilevel implicit methods, Dirichlet’s problem, Neumann problem, mixed boundary value problem to solve the heat conduction equation. Explicit methods, implicit methods for one space dimension and two space dimensions are used to solve the hyperbolic equation. In mathematics, a hyperbolic partial differential equation has a well-posed initial value problem. The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Finite Element methods are included in Unit-V and VI. The finite element method (FEM) is anumerical technique for finding approximate solutions to boundary value problems for differential equations. Unit-V contains heat conduction equation of heat transfer, Governing differential equation for heat conduction, Formulation of finite element method for heat conduction. Unit -VI contains Galerkin’s methods for 1D, 2D, 3D heat conduction, Transient heat conduction problems solving by Finite element method. 58 
