math264-note-2007-1

math264-note-2007-1 - INTRODUCTION OF PARTIAL DIFFERENTIAL...

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INTRODUCTION OF PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES PROF. JIAN-JUN XU (OCTOBER, 2007) Date : October. 2007. 0
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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 ) .
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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 0 ( A ) X, where ( A ) = ˆ a b b c ! ; X = ˆ x y ! ; X 0 = ( 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 0 y 0 ) 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 0 y 0 ! ;
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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 0 = Y 0 ( Q ) 0 . Therefore, in the new Cartesian coordinate system, the quadratic form can be written as I ( x 0 y 0 ) = Y 0 ( B ) Y where the matrix ( B ) is given by the formula. ( B ) = ( Q ) 0 ( A )( Q ) . With the above coordinate system transformation, the quadratic curve I ( x, y ) = 1, then is written in the form I ( x 0 , y 0 ) = 1 in the new system. By the theory of linear algebra, it is well known that in terms of a properly chosen matrix ( Q ), the matrix ( B ) can be reduced into a diagonal matrix, so that the quadratic curve I ( x, y ) = 1 can be written in the old system can be reduced to the standard form: I ( x 0 , y 0 ) = λ 1 x 0 2 + λ 2 y 0 2 = 1 . The numbers ( λ 1 , λ 2 ) are subject to the formula: det( A ) = ac - b 2 = det( B ) = λ 1 λ 2 .
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