SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures 37-39 KFUPM Read 29.1-29.2 & 30.1-30.4 CISE301_Topic9 KFUPM

Lecture 37 Partial Differential Equations Partial Differential Equations (PDEs). What is a PDE? Examples of Important PDEs. Classification of PDEs. CISE301_Topic9 KFUPM

Partial Differential Equations A partial differential equation (PDE) is an equation that involves an unknown function and its partial derivatives. CISE301_Topic9 KFUPM

Notation CISE301_Topic9 KFUPM

Linear PDE Classification CISE301_Topic9 KFUPM

Representing the Solution of a PDE (Two Independent Variables) Three main ways to represent the solution T=5.2 t1 T=3.5 x1 Different curves are used for different values of one of the independent variable Three dimensional plot of the function T(x,t) The axis represent the independent variables. The value of the function is displayed at grid points CISE301_Topic9 KFUPM

Different curve is used for each value of t Heat Equation Different curve is used for each value of t ice ice Temperature Temperature at different x at t=0 x Thin metal rod insulated everywhere except at the edges. At t =0 the rod is placed in ice Position x Temperature at different x at t=h CISE301_Topic9 KFUPM

Heat Equation Time t ice ice x t1 x1 Temperature T(x,t) Position x CISE301_Topic9 KFUPM

Linear Second Order PDEs Classification CISE301_Topic9 KFUPM

Linear Second Order PDE Examples (Classification) CISE301_Topic9 KFUPM

Classification of PDEs Linear Second order PDEs are important sets of equations that are used to model many systems in many different fields of science and engineering. Classification is important because: Each category relates to specific engineering problems. Different approaches are used to solve these categories. CISE301_Topic9 KFUPM

Examples of PDEs PDEs are used to model many systems in many different fields of science and engineering. Important Examples: Wave Equation Heat Equation Laplace Equation Biharmonic Equation CISE301_Topic9 KFUPM

Heat Equation The function u(x,y,z,t) is used to represent the temperature at time t in a physical body at a point with coordinates (x,y,z) . CISE301_Topic9 KFUPM

Simpler Heat Equation x u(x,t) is used to represent the temperature at time t at the point x of the thin rod. CISE301_Topic9 KFUPM

Wave Equation The function u(x,y,z,t) is used to represent the displacement at time t of a particle whose position at rest is (x,y,z) . Used to model movement of 3D elastic body. CISE301_Topic9 KFUPM

Laplace Equation Used to describe the steady state distribution of heat in a body. Also used to describe the steady state distribution of electrical charge in a body. CISE301_Topic9 KFUPM

Biharmonic Equation Used in the study of elastic stress. CISE301_Topic9 KFUPM

Boundary Conditions for PDEs To uniquely specify a solution to the PDE, a set of boundary conditions are needed. Both regular and irregular boundaries are possible. t region of interest x 1 CISE301_Topic9 KFUPM

The Solution Methods for PDEs Analytic solutions are possible for simple and special (idealized) cases only. To make use of the nature of the equations, different methods are used to solve different classes of PDEs. The methods discussed here are based on the finite difference technique. CISE301_Topic9 KFUPM

Lecture 38 Parabolic Equations Heat Conduction Equation Explicit Method Implicit Method Cranks Nicolson Method CISE301_Topic9 KFUPM

Parabolic Equations CISE301_Topic9 KFUPM

Parabolic Problems ice ice x CISE301_Topic9 KFUPM

First Order Partial Derivative Finite Difference Forward Difference Method Backward Difference Method Central Difference Method CISE301_Topic9 KFUPM

Finite Difference Methods CISE301_Topic9 KFUPM

Finite Difference Methods New Notation Superscript for t-axis and Subscript for x-axis Til-1=Ti,j-1=T(x,t-∆t) CISE301_Topic9 KFUPM

Solution of the PDEs t x CISE301_Topic9 KFUPM

Solution of the Heat Equation Two solutions to the Parabolic Equation (Heat Equation) will be presented: 1. Explicit Method: Simple, Stability Problems. 2. Crank-Nicolson Method: Involves the solution of a Tridiagonal system of equations, Stable. CISE301_Topic9 KFUPM

Explicit Method CISE301_Topic9 KFUPM

Explicit Method How Do We Compute? u(x,t+k) u(x-h,t) u(x,t) u(x+h,t) CISE301_Topic9 KFUPM

Explicit Method How Do We Compute? CISE301_Topic9 KFUPM

Explicit Method CISE301_Topic9 KFUPM

Crank-Nicolson Method CISE301_Topic9 KFUPM

Explicit Method How Do We Compute? u(x-h,t) u(x,t) u(x+h,t) u(x,t - k) CISE301_Topic9 KFUPM

Crank-Nicolson Method CISE301_Topic9 KFUPM

Crank-Nicolson Method CISE301_Topic9 KFUPM

Examples Explicit method to solve Parabolic PDEs. Cranks-Nicholson Method. CISE301_Topic9 KFUPM

Heat Equation ice ice x CISE301_Topic9 KFUPM

Example 1 CISE301_Topic9 KFUPM

Example 1 (Cont.) CISE301_Topic9 KFUPM

Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Remarks on Example 1 CISE301_Topic9 KFUPM

Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 2 CISE301_Topic9 KFUPM

Example 2 Crank-Nicolson Method CISE301_Topic9 KFUPM

Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 2 Crank-Nicolson Method CISE301_Topic9 KFUPM

Example 2 Second Row t=1.0 t=0.75 u1 u2 u3 t=0.5 t=0.25 t=1.0 t=0.75 u1 u2 u3 t=0.5 t=0.25 0.2115 0.2991 0.2115 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=0.25 x=0.5 x=0.75 x=1.0 CISE301_Topic9 KFUPM

Example 2 The process is continued until the values of u(x,t) on the desired grid are computed. CISE301_Topic9 KFUPM

Remarks The Explicit Method: One needs to select small k to ensure stability. Computation per point is very simple but many points are needed. Cranks Nicolson: Requires the solution of a Tridiagonal system. Stable (Larger k can be used). CISE301_Topic9 KFUPM

Lecture 39 Elliptic Equations Laplace Equation Solution CISE301_Topic9 KFUPM

Elliptic Equations CISE301_Topic9 KFUPM

Laplace Equation Laplace equation appears in several engineering problems such as: Studying the steady state distribution of heat in a body. Studying the steady state distribution of electrical charge in a body. CISE301_Topic9 KFUPM

Laplace Equation Temperature is a function of the position (x and y) When no heat source is available f(x,y)=0 CISE301_Topic9 KFUPM

Solution Technique A grid is used to divide the region of interest. Since the PDE is satisfied at each point in the area, it must be satisfied at each point of the grid. A finite difference approximation is obtained at each grid point. CISE301_Topic9 KFUPM

Solution Technique CISE301_Topic9 KFUPM

Solution Technique CISE301_Topic9 KFUPM

Solution Technique CISE301_Topic9 KFUPM

Example It is required to determine the steady state temperature at all points of a heated sheet of metal. The edges of the sheet are kept at a constant temperature: 100, 50, 0, and 75 degrees. 100 75 50 The sheet is divided to 5X5 grids. CISE301_Topic9 KFUPM

Example Known To be determined CISE301_Topic9 KFUPM

First Equation Known To be determined CISE301_Topic9 KFUPM

Another Equation Known To be determined CISE301_Topic9 KFUPM

Solution The Rest of the Equations CISE301_Topic9 KFUPM

Convergence and Stability of the Solution The solutions converge means that the solution obtained using the finite difference method approaches the true solution as the steps approach zero. Stability: An algorithm is stable if the errors at each stage of the computation are not magnified as the computation progresses. CISE301_Topic9 KFUPM