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1. PADMAVANI ARTS AND SCIENCE COLLEGE FOR WOMENSALEM-11PG &RESEARCH DEPARTMENT OF MATHEMATICSMs.P.ELANGOMATHI M.sc., M.Phil.,M.Ed., SUB: PARTIAL DIFFERENTIAL EQUATIONS

2. UNIT 1- second order Differential equation ORIGIN OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATION:Consider the function z=f(u)+g(v)+w → (1)Where f and g are arbitrary functions of u and v respectively and u , v and w are functions of x and y. We write

3. P=∂z/∂x , q=∂z/∂y , r=∂/∂x(∂z/∂x) , s=∂/∂x(∂z/∂y) , t=∂/∂y(∂z/∂y)Differentiating these terms with respect to x &y respectively. And simplifying the equations we get , Rr+Ss+Tt+Pp+Qq=W → (2)Where R,S,T,P,Q and W are known functions of x &y . Therefore equation (1) is a solution of the second order differential equation (2) which is a particular type of the equation and contains dependent variable z. LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS:An equation of the form F(D , D’)z = f(x ,y ) Where F(D,D’) is a differential operator.

4. D=∂/∂x , D’=∂/∂y is called a linear partial differential equation with constant coefficients. DEFINITIONS: REDUCIBLE . IRREDUCIBLE. METHODS OF SOLVING LINEAR PDE: SOLUTION OF REDUCIBLE EQUATION: SOLUTION OF IRREDUCIBLE EQUATIONS WITH CONSTANT COEFFICIENTS: RULES FOR FINDING COMPLEMENTARY FUNCTION: RULES FOR FINDING PARTICULAR INTEGRAL: CLASSIFICATION OF SECOND ORDER PDE

5. A second order partial differential equation which is linear with respect to second order partial derivatives that is, r, s &t is said to be quasi linear PDE of second order. Rr+Ss+Tt+f(x ,y ,z, p, q)=0 →→(1)The equation is said to be elliptic if (<0) parabolic if (=0) hyperbolic if (>0)CANONICAL FORMS: The order to reduce the PDE (1) to a canonical form we apply the transformation . ζ=ζ(x ,y) , η=η(x, y).

6. Such that the function ζ and η are continuously differentiable and the jacobian J=∂(ζ, η)/ ∂( x, y). ADJOINT OPERATORS:Let Lu=Φ where L is the differential operator. if the operator L=L* then L is called a self adjoint operator. RIEMANN ‘S MATHOD : we known that any linear hyperbolic partial differential equation can be written in the form

7. ∂/∂x (∂z/∂y) +a ∂z/∂x + b ∂z/∂y + c z= f(x, y)Where a, b, c are functions of x &y . L(z)= f(x, y).Then w to satisfy the following conditions M(w)=0 ∂w/∂y=a w, on x=ζ ∂w/∂x=b w, on y=η [w]p= 1

8. Unit 2- elliptic differential equationOCCURRENCE OF THE LAPLACE AND POISSAN EQUATIONS:1.1. DERIVATION OF LAPLACE EQUATION: Let two particles of masses m1& m2 be situated at the point P& Q at the distance r apart. According to newton’s law of gravitation , the magnitude of the force is directly proportional to the product of their masses and inversely

9. Proportional to the square of the distance between them and is given by F=G( m1.m2/r . r)Where G is the gravitational constant. It can be verified that ▼(▼.V)=0 which is the laplace equation. 2. DERIVATION OF POISSAN EQUATION: let S be a closed surface consisting a particle of masses m1, m2 ,m3, m4, ……….mn . Let Q be any point on S and Σmi= M be the total masses inside S. let g1, g2, ……g n be the gravitational fields at Q due to m1, m2, m3, …….. mn respectively. By using gauss law , we get ▼(▼.V)= - 4ΠGρ is called poisson’s equation.

10. 3. BOUNDARY VALUE PROBLEMS:If f belongs to C and is prescribed on the boundary C of some finite region R, the program of determining a function Φ( x, y, z) such that ▼(▼Φ)=0 within R and satisfying Φ=f on C is called the boundary value problem of first kind or dirichlet problem. The second kind of BVP is to determine the function Φ(x, y, z) so that ▼(▼Φ)=0 within R while∂Φ/∂n is specified at every point of C, where ∂Φ/∂n is the normal derivative of Φ . This problem is called the neumann problem.

11. The third type of BVP is concerned with the determination of the function Φ(x, y, z) such that ▼(▼Φ)=0 within R , while a boundary condition of the form ∂Φ/∂n +hΦ=f , where h≥0 is specified at every point of the boundary C. this is called a mixed boundary value problem or churchill,’s problem. 3. SEPARATIONOF VARIABLES METHOD: consider a two laplace equation ▼(▼u)=∂/∂x(∂u/∂x)+∂/∂y(∂u/∂y)= 0We assume that u(x, y)= X(x)Y(y).

12. 4. LAPLACE EQUATION IN CYLINDRICAL COORDINATES:The laplace equation in cylindrical coordinates is given by ▼(▼u)=∂/∂r (∂u/∂r)+1/r (∂u/∂r)+1/r . r ∂/∂θ(∂u/∂θ)+∂/∂z(∂u/∂z). Its solution form u(r,θ,z).5. LAPLACE EQUATION IN SPHERICAL COORDINATES:The laplace equation in spherical coordinates is given by ▼(▼u)=∂/∂r((r.r)∂u/∂r)+1/sinθ∂/∂θ(sinθ∂u/∂θ)+1/sin^2θ∂/∂Φ(∂u/∂Φ)Its solution form u(r,θ,Φ).6. INTERIOR & EXTERIOR DIRICHLET PROBLEM FOR A CIRCLE:7. INTERIOR & EXTERIORPROBLEM FOR A SPHERE:

13. UNIT 3- parabolic differentiol equationOCCURRENCE AND DERIVATION OF THE DIFFUSION EQUATION:Let T(x, y, z, t) be the temperature at the point P(x, y, z) at time t . Then ∂T/∂t (r, t)=K▼(▼ T(r, t)).Is called the diffusion equation. . BOUNDARY CONDITIONS: Boundary condition 1 ( homogeneous boundary condition) Boundary condition 2 ( neumann condition) Boundary condition 3 ( robin’s condition)

14. 3. SEPARATION OF VARIABLES METHOD:We consider the one dimensional heat conduction equation ∂T/∂t= K ∂/∂x(∂T/∂x)Its solution is T( x, t)= X(x).Y(t).4. DIFFUSION EQUATION IN CYLINDRICAL COORDINATES: consider a three dimensional diffusion equation ∂T/∂t=K ▼(▼T).Its solution is T(r, θ, z)= R(r).θ(θ). Z(z).Φ(t).DIFFUSION EQUATION IN SPHE3RICAL COORDINATES:In the spherical polar coordinates the heat conduction equation ∂T/∂t=K ▼(▼T).Its solution is T=R(r).θ(θ).Φ(Φ).Ψ(t).

15. Unit 4- hyperbolic differentiol equation OCCURRENCE OF THE WAVE EQUATION: One of the most important and typical homogeneous hyperbolic differential equations is the wave equation of the form ∂/∂t(∂u/∂t)=(c . c) ▼(▼u).The solution of wave equation is called wave function. DERIVATION OF ONE – DIMENSIONAL WAVE EQUATION:

16. We assume the following : the motion takes place in one plane only and in this plane each particle moves in a direction perpendicular to the equilibrium position of the string. the tension T in string is constant . the gravitational force is neglected as compared with tension T of the string . the slope of the deflection curve is small.Hence a string of wave equation is ∂/∂t(∂y/∂t)=(c . c) ∂/∂x(∂y/∂x). where (c . c)= λ/ρ.

17. REDUCTION OF ONE DIMENSIONAL WAVE EQUATION TO CANONICAL FORM AND ITS SOLUTION: consider the one dimensional wave equation Utt =(c . c) UxxIts solution is u(x, t)= Φ[kx-wt]+Φ[kx+wt]. D’ALEMBERT SOLUTION OF ONE DIMENSIONAL WAVE EQUATION :Consider the initial value problem of cauchy type described as Utt=(c . c) UxxSubject to initial conditions u(x, 0)=η(x) ; Ut(x, 0)= v(x).Its solution is u(x, t)= ½[ η(x-ct)+η(x+ct)].

18. SEPERATION OF VARIABLES METHOD: PERIODIC SOLUTIONS: CYLINDRICAL COORDINATES : SPHERICAL POLAR COORDINATES: DUHAMEL’S PRINCIPLE FOR WAVE EQUATION:The partial differential equation is Vtt(X, t)- (c .c )▼(▼v (X, t)). With the conditions v(X,0,λ)=0; Vt(X,0,λ)=F(X,λ) at t=λ.

19. Unit 5- laplace transformsLAPLACE TRANSFORMS: DEFINITIONS : INVERSE LAPLACE TRANSFORM INTEGRAL:SOLUTION FOR PARTIAL DIFFERENTIAL EQUATIONS; DIFFUSION EQUATION:Use laplace transform method, k Ut= Utt, 0<x<l; 0<t<∞

20. WAVE EQUATION :Using laplace transform method Uxx= 1/ (c.c) Utt- cos wt, by condition 0≤x≤∞ ; 0≤t≤ ∞. FOURIER TRANSFORMS AND THEIR APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATION S: LAPLACE EQUATION. DIFFUSION EQUATION . WAVE EQUATION.

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