PARTIAL DIFFERENTIAL EQUATIONS - PowerPoint Presentation

Slide1

PARTIAL DIFFERENTIAL EQUATIONSSlide2

Formation of Partial Differential equations

Partial Differential Equation can be formed either by elimination of arbitrary constants or by the elimination of arbitrary functions from a relation involving three or more variables . SOLVED PROBLEMS1.Eliminate two arbitrary constants a and b from here R is known constant . Slide3

solution

Differentiating both sides with respect to x and y

(OR) Find the differential equation of all spheres

of fixed radius having their centers in x y- plane.Slide4

By substituting all these values in (1)

orSlide5

2. Find the partial Differential Equation by eliminating

arbitrary functions from

SOLUTIONSlide6

BySlide7

3.Find Partial Differential Equation

by eliminating two arbitrary functions from

SOLUTION

Differentiating both sides with respect to x and ySlide8

Again d . w .r. to x and yin equation (2)and(3

)Slide9
Slide10

Different Integrals of Partial Differential Equation

1. Complete Integral (solution) Let be the Partial Differential Equation.

The complete integral of equation (1) is given

by

Slide11

2. Particular solution

A solution obtained by giving particular values tothe arbitrary constants in a complete integral iscalled particular solution .

3.Singular solution

The

eliminant

of a , b between

when it exists , is called singular solution

Slide12

4. General solution

In equation (2) assume an arbitrary relation of the form . Then (2) becomes

Differentiating (2) with respect to a,

The

eliminant

of (3) and (4) if exists,

is called general solutionSlide13

Standard types of first order equationsTYPE-I The Partial Differential equation of the form

has solution

with

TYPE-II

The Partial Differential Equation of the form

is called

Clairaut’s

form

of

pde

,

it’s solution is given by Slide14

TYPE-III

If the pde

is given by

then assume that Slide15

The given

pde

can be written as

.

And also this can

be integrated to get solution Slide16

TYPE-IV

The pde of the form can be solved by assuming

Integrate the above equation to get solutionSlide17

SOLVED PROBLEMS

1.Solve the pde

and find the complete

and singular solutions

Solution

Complete solution is given by

withSlide18

d.w.r.to. a and c then

Which is not possible

Hence there is no singular solution

2.Solve the

pde

and find the complete, general and singular solutionsSlide19

Solution

The complete solution is given by

withSlide20

no singular solution

To get general solution assume that

From

eq

(1)Slide21

Eliminate from (2) and (3) to get general solution

3.Solve the

pde

and find the complete and singular solutions

Solution

The

pde

is in

Clairaut’s

form Slide22

complete solution of (1) is

d.w.r.to “a” and “b” Slide23

From (3)Slide24

is required singular solutionSlide25

4.Solve the

pde

Solution

pde

Complete solution of above

pde

is

5.Solve the

pde

Solution

Assume thatSlide26

From given

pdeSlide27

Integrating on both sidesSlide28

6

. Solve the pde

Solution

Assume

Substituting in given equationSlide29

Integrating on both sides

7.Solve

pde

(or)

SolutionSlide30

Assume that

Integrating on both sides Slide31

8. Solve the equation

Solution

integratingSlide32

Equations reducible to the standard forms

(i

)If and occur in the

pde

as in

Or in

Case (a) Put and

if ; Slide33

where

Then reduces to

Similarly reduces toSlide34

case(b)

If or put

(ii)If and occur in

pde

as in

Or inSlide35

Case(a) Put if

where

Given

pde

reduces to

andSlide36

Case(b) if

Solved Problems

1.Solve

Solution Slide37

whereSlide38

(1)becomesSlide39

2. Solve the

pde

SOLUTIONSlide40

Eq

(1) becomesSlide41

Lagrange’s Linear Equation

Def: The linear partial

differenfial

equation

of first order is called as Lagrange’s linear Equation.

This

eq

is of the form

Where and are functions

x,y

and z

The general solution of the partial differential equation is

Where is arbitrary function of

and Slide42

Here and are independent solutions

of the

auxilary

equations

Solved problems

1.Find the general solution of

Solution

auxilary

equations areSlide43

Integrating on both sides

Integrating on both sidesSlide44

The general solution is given by

2.solve

solution

Auxiliary equations are given bySlide45

Integrating on both sidesSlide46

Integrating on both sidesSlide47

The general solution is given by

HOMOGENEOUS LINEAR PDE WITH CONSTANT COEFFICIENTS

Equations in which partial derivatives occurring are all of same order (with degree one ) and the coefficients are constants ,such equations are called homogeneous linear PDE with constant coefficientSlide48

Assume that

then order linear homogeneous equation is given by

orSlide49

The complete solution of equation (1) consists of two parts ,the complementary function and particular integral.

The complementary function is complete solution of equation of

Rules to find complementary function

Consider the equation

orSlide50

The auxiliary equation for (A.E) is given by

And by giving

The A.E becomes

Case 1

If the equation(3) has two distinct roots

The complete solution of (2) is given bySlide51

Case 2

If the equation(3) has two equal roots i.e

The complete solution of (2) is given by

Rules to find the particular Integral

Consider the equationSlide52

Particular Integral (P.I)

Case 1

If

then P.I=

If and is

factor of then Slide53

Case 2

P.I

P.I

If and is

factor of

then P.ISlide54

Case 3

P.I

Expand

in ascending powers of

or and operating on term by term.

Case 4

when is any function of x and y.

P.I=

Slide55

Solved problems

1.Find the solution of

pde

Solution

The Auxiliary equation is given by

Where ‘c’ is replaced by after integration

Here

is factor ofSlide56

Solution

The Auxiliary equation is given by By taking

Complete solution

2

. Solve the

pde

Solution

The Auxiliary equation is given bySlide57

3

. Solve the pde

Solution

the A.E is given bySlide58

4

. Find the solution of pde

Solution

Complete solution =

Complementary Function + Particular Integral

The A.E is given bySlide59

Complete solution Slide60

5

.Solve

SolutionSlide61
Slide62

6

.Solve

SolutionSlide63

7

.Solve

SolutionSlide64
Slide65
Slide66

7

.Solve

Solution

A.E isSlide67
Slide68
Slide69

Non Homogeneous Linear PDES

If in the equation

the polynomial expression is 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

ExSlide70

If the non homogeneous equation is of the form

1

.Solve

SolutionSlide71
Slide72

AssignmentSlide73

1.Find the differential equation of all spheres of fixed radius having centre in

xy-plane.2.Solve the

pde

z=ax

3

+by

3

by eliminating the arbitrary constants.

3.Solve

the

pde

z=f(x

2

-y

2

)

by eliminating the arbitrary

constants.

4.solveSlide74

5

.(D²+2DD+D

²

)z=exp(2x+3y)

6.4r-4s+t=6log(x+2y

)

7.find

the general

solution of differential equation (D²+D



+4)z=exp(4x-y)

Slide75

TEST

NOTE:- DO ANY TWOSlide76

1.(D-D

-1)(D-D-2)Z=EXP(2X-Y)2.SOLVE(D3+D2D

-DD



2

-D

3)Z=EXP(X)COS2Y

3.SOLVE (X-Y)p+(X+Y)q=2XZ