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partial differential equations

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• MRCE, B.Tech ECE 1st year
• MRCE, B.Tech ECE 1st year
• ### Partial

1. 1. APPLICATION OF PARTIAL DIFFERENTIAL EQUATIONS <ul><li>Presented By: </li></ul><ul><li>Rahul Sharma </li></ul><ul><li>Ravinder </li></ul><ul><li>Tamesh </li></ul><ul><li>Tejaasvi Bhogra </li></ul>MRCE, B.Tech ECE 1st year
2. 2. INTRODUCTION <ul><li>These equations are usually classified according to their mathematical form. </li></ul><ul><li>Differential equations involving two or more independent variables are called partial differential equations. </li></ul><ul><li>These equations may have only boundary conditions, in which they are referred to as Boundary Value Problems (BVP) or steady-state equations. </li></ul>MRCE, B.Tech ECE 1st year
3. 3. CLASSIFICATION <ul><li>Wave Equation : </li></ul><ul><li>1-D Heat flow : </li></ul><ul><li>2-D Heat flow: </li></ul><ul><li>Radio Equations: </li></ul>MRCE, B.Tech ECE 1st year
4. 4. Methods of Separation Of Variables <ul><li>Assumption </li></ul><ul><li>Dependent Variable is the product of 2 functions, each involving only one of the independent Variables. </li></ul><ul><li>Outcome : 2 Ordinary Differential Equations are Formed. </li></ul>MRCE, B.Tech ECE 1st year
5. 5. Equation of Vibrating String OR 1D Wave Equation <ul><li>The boundary conditions to be satisfied by the Equation are : </li></ul><ul><li>y=0 ,when x=0 </li></ul><ul><li>Y=0 , when x =1 </li></ul><ul><li>[ These should be satisfied by every value of ‘t’ ] </li></ul>MRCE, B.Tech ECE 1st year
6. 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. 7. SOLUTION OF THE HEAT EQUATION MRCE, B.Tech ECE 1st year
8. 8. 2 D Heat Flow <ul><li>Note 1 : in the steady state,u is independent of t,so that du/dt = 0 </li></ul><ul><li>d2u/dx2 + d2u/dy2 = 0 </li></ul><ul><li>Which is Laplace’s Equation in 2-D </li></ul>MRCE, B.Tech ECE 1st year
9. 9. 2-D Heat Flow <ul><li>(d2u/dx2 + d2u/dy2 +d2u/dz2 ) = du/dt </li></ul><ul><li>In Steady state,it reduces to </li></ul><ul><li>(d2u/dx2 + d2u/dy2 +d2u/dz2 ) = 0 </li></ul><ul><li>Which is Laplace’s Equation in 3-D </li></ul>MRCE, B.Tech ECE 1st year
10. 10. Solution Of Laplace’s Equation in 2D MRCE, B.Tech ECE 1st year
11. 11. THANK YOU MRCE, B.Tech ECE 1st year