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**Unformatted text preview: **INTRODUCTION OF PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES PROF. JIAN-JUN XU (OCTOBER, 2007) Date : October. 2007. INTRODUCTION OF PDE’S AND FOURIER SERIES 1 1. INTRODUCTION 1.1. General Form of P.D.E. F ( x,y,z,t, Φ x , Φ y , Φ z , Φ xx , Φ yy , Φ tt ... ) = 0 . (1.1) where • ( x,y,z ) : Space Coordinates • t : Time • Φ: Vector function. 1.2. Definitions. 1.2.1. Order of PDE – n. • if Φ( x,y,z,t ) is scalar, n is the highest order n of the derivatives; • if Φ( x,y,z,t ) is vector, n = m × n d , where m : number of the components; n d : the highest order. 1.2.2. Solution of PDE. Φ( x,y,z,t ) has all necessary derivatives and satisfied the equation (1.1) and all conditions. Remark 1.2.1 . : Week solution; discontinuous solutions 1.3. Linear and Non-linear Equations. Examples : (1) Linear A ( x,y,z ) u x + B ( x,y,z ) u y = C ( x,y,z ) (2) Quasi-linear A ( x,u ) u x + B ( x,u ) u y = C ( x,u ) (3) Semi-linear A ( x,y,z ) u x + B ( x,y,z ) u y = C ( x,y,z,u ) (4) Non-linear A ( x,y,z ) u 2 x + B ( x,y,z ) u 1 2 y = W. 1.4. General Solution. Some PDE’s have the general solution. Examples : (1) Φ xy ( x,y ) = 0 ( n = 2) ⇒ Φ( x,y ) = f ( x ) + g ( y ) . 2 PROF. JIAN-JUN XU (OCTOBER, 2007) (2) 2 ∂u ∂x- ∂u ∂y = 0 ( n = 1) ⇒ u = f ( x + 2 y ) . where f ( s ) is arbitrary function. (3) ∂ 2 u ∂x 2- ∂ 2 u ∂y 2 = 0 , n = 2 ∂ ∂x- ∂ ∂y ¶ ∂ ∂x + ∂ ∂y ¶ u = 0 let ( s = x + y τ = x- y ⇒ 4 ∂ ∂s ¶ ∂ ∂τ ¶ = 0 ⇒ u = f ( τ ) + g ( s ) , where f,g are arbitrary functions. In general, P.D.E has no general solution only has particular solution under some conditions. 1.5. Classification of 2-nd Linear PDE’s. The general form of 2-nd Linear PDE’s: ‡ a ∂ 2 ∂x 2 + 2 b ∂ 2 ∂x∂y + c ∂ 2 ∂y 2 + L . O . T · Φ( x,y ) = 0 where the coefficients a,b,c are functions of ( x,y ). As comparison, one may consider the quadratic form: I ( x,y ) = ax 2 + 2 bxy + cy 2 = X ( A ) X, where ( A ) = ˆ a b b c ! ; X = ˆ x y ! ; X = ( x,y ) . If one rotates the Cartesian coordinate system around the origin, as a result, the coordinates ( x,y ) in the old system of a fixed point P ( x,y ) will transformed to ( x y ) in the new system, such that X = ( Q ) Y, where ( Q ) = ˆ q 11 q 12 q 21 q 22 ! ; is a normal-orthogonal transformation matrix, while Y = ˆ x y ! ; INTRODUCTION OF PDE’S AND FOURIER SERIES 3 Note that from the above relationship, one derives that the transport of vector X is given by X = Y ( Q ) . Therefore, in the new Cartesian coordinate system, the quadratic form can be written as I ( x y ) = Y ( B ) Y where the matrix ( B ) is given by the formula. ( B ) = ( Q ) ( A )( Q ) ....

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