This document discusses the application of partial differential equations. It begins by classifying partial differential equations according to their mathematical form as either boundary value problems or steady-state equations. Some common partial differential equations are then presented, including the wave equation, heat equations, and Laplace's equation. Solution methods like separation of variables are introduced. Specific examples of the 1D wave equation and 1D heat equation are then covered. Finally, the document discusses the Laplace equation in 2D and 3D.

1. APPLICATION OF PARTIAL DIFFERENTIAL EQUATIONS Presented By: Rahul Sharma Ravinder Tamesh Tejaasvi Bhogra MRCE, B.Tech ECE 1st year

2. INTRODUCTION These equations are usually classified according to their mathematical form. Differential equations involving two or more independent variables are called partial differential equations. These equations may have only boundary conditions, in which they are referred to as Boundary Value Problems (BVP) or steady-state equations. MRCE, B.Tech ECE 1st year

3. CLASSIFICATION Wave Equation : 1-D Heat flow : 2-D Heat flow: Radio Equations: MRCE, B.Tech ECE 1st year

4. Methods of Separation Of Variables Assumption Dependent Variable is the product of 2 functions, each involving only one of the independent Variables. Outcome : 2 Ordinary Differential Equations are Formed. MRCE, B.Tech ECE 1st year

5. Equation of Vibrating String OR 1D Wave Equation The boundary conditions to be satisfied by the Equation are : y=0 ,when x=0 Y=0 , when x =1 [ These should be satisfied by every value of ‘t’ ] MRCE, B.Tech ECE 1st year

6. 1 Dimensional Heat Flow = K/= K/s ρ which is known as Diffusivity of the material of the bar . Where , S = specific Heat ρ = density of material & K = Conductivity. MRCE, B.Tech ECE 1st year

7. SOLUTION OF THE HEAT EQUATION MRCE, B.Tech ECE 1st year

8. 2 D Heat Flow Note 1 : in the steady state,u is independent of t,so that du/dt = 0 d2u/dx2 + d2u/dy2 = 0 Which is Laplace’s Equation in 2-D MRCE, B.Tech ECE 1st year

9. 2-D Heat Flow (d2u/dx2 + d2u/dy2 +d2u/dz2 ) = du/dt In Steady state,it reduces to (d2u/dx2 + d2u/dy2 +d2u/dz2 ) = 0 Which is Laplace’s Equation in 3-D MRCE, B.Tech ECE 1st year

10. Solution Of Laplace’s Equation in 2D MRCE, B.Tech ECE 1st year

11. THANK YOU MRCE, B.Tech ECE 1st year