Name :- Vrajesh shah(150410116108)Sub :- Advanced engineering mathematics Topic:- Higherorder Non Homogeneous Partial Differential EquationsDepartment :-IT

SARDAR VALLABHBHAI PATEL INSTITUTE OF TECHNOLOGY

Partial DifferentiationDefinition :-

A partial differential equation is an equation involving a function of two or more variables and some of its partial derivatives. Therefore a partial differential equation contains one dependent variable and more than one independent variable. Here z will be taken as the dependent variable and x and y the independent variable so that .

We will use the following standard notations to denote the partial derivatives.

yxfz , .

,, qyzp

xz

t

yzs

yxzr

xz

2

22

2

2

,,

Solution to non homogeneous partial differential equation

General Form of 2nd order Non-Homogeneous Partial differential equations :-

Where

( F (D , D’) Z = f ( , y ) Solution is given by Z = Complimentary Function (C.F) + Particular Integral (P.I) Complimentary Function (From L.H.S) Particular Integral (From R.H.S)

Non Homogeneous Linear PDES

If in the equation

the polynomial expressionis not homogeneous, then (1) is a non- homogeneous linear partial differential equation

Complete Solution = Complementary Function + Particular Integral

To find C.F., factorize into factors of the form

Ex

¿

(𝐷2+3𝐷+𝐷′−4𝐷 ′ )𝑍=𝑒2 𝑥+3 𝑦

(𝐷−𝑚𝐷′−𝐶 )

If the non homogeneous equation is of the form

)()(.

),())((

21

2211

21 xmyexmyeFC

yxFzcDmDcDmDxcxc

1.Solve

Solution:- )1(),( 2 DDDDDDDDDf

)()(. 21 yxyeFC x

 

6.5.125.4.34.31231

......)1()1(1

)1(11.

654432

2

22

222

21

22

2

xxxxxxD

xDDx

DDx

D

xDD

DDDDDxIP

2.Solve 4)32)(1( zDDDD

Solution

34)2()( 1

31 xyexyez xx

Case II) :- Roots are repeated

m1=m2=m

)()(. 21 xmyxexmyeFC xcxc

Rules for finding Particular Integral F ( D , D’ ) Z = f ( , y )

Case I :- f ( , y ) =

P.I = , P.I = ; f ( a , b ) ≠ 0

Case II :-

P.I = P.I = ,

Case III :- P.I = If m<n then expansion is in powers of If m>n then expansion is in powers of Use :- 1.

2. ;

Case IV (General Rule) :- (Rule for failure case )

After integration , Substitute c = y + mx

Example:-11) The Auxiliary equation is given by

m = -1 , -1Roots are repeated C.F = P.I = P.I = P.I = P.I= Solution is Z = C.F + P.I Z =

Example :- 22) The Auxiliary equation is given by

m(m-1)=0

Roots are real and distinct

m=0, 1 ----ROOTS

C.F =

P.I = P.I= P.I = P.I= P.I= P.I= P.I=Solution is Z = C.F + P.I

Z =

PDEs are used to model many systems in many different fields of science and engineering.

Important Examples:

Laplace Equation Heat Equation Wave Equation

Application of pde:

Laplace Equation is 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.

LAPLACE EQUATION:

0),,(),,(),,(2

2

2

2

2

2

z

zyxuy

zyxux

zyxu

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)

is the thermal diffusivity. It is sufficient to consider the case = 1.

HEAT EQUATION:

2

2

2

2

2

2

),,,(zu

yu

xu

ttzyxu

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) .

The constant c represents the propagation speed of the wave.

WAVE EQUATION:

2

2

2

2

2

22

2

2

),,,(zu

yu

xuc

ttzyxu

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation instochastic partial differential equations.

APPLICATIONS

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