Chapter 6 Partial Differential Equations

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42 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS or a (ϕ 1x + iϕ 2x )+ ( b + √ ) ac − b 2 )(ϕ 1y + iϕ 2y ) =0. Note that, since we have assumed that a, b, c are real the solution ψ of aψ x + (b − i √ ) ac − b 2 ψ y =0, is given by ψ = ϕ = ϕ 1 − iϕ 2 . To argue that these complex equations are solvable we collecting real and imaginary parts in the equation for ϕ to obtain a first order linear system of partial differential equations or aϕ 1x + bϕ 1y − √ (ac − b 2 ) ϕ 2y =0 aϕ 2x + bϕ 2y + √ (ac − b 2 ) ϕ 1y =0. ( ) ∂ ϕ1 = 1 ( −b ∂x ϕ 2 a − √ (ac − b 2 ) If we prescribe any Cauchy initial data ( ϕ1 ϕ 2 ) (0,y)= √ ) (ac − b2 ) ∂ −b ∂y ( ) g1 (y) g 2 (y) ( ϕ1 ϕ 2 ) . this system has a unique solution by the Cauchy-Kovalevski Theorem. We now introduce the real transformations and note that ξ = ϕ 1 =Reϕ, η = ϕ 2 =Imϕ ( ) ϕ1x ϕ det J = det 1y ϕ 2x ϕ 2y ⎛ = 1 a det ⎝ −bϕ 1y + √ ⎞ (ac − b 2 )ϕ 2y ϕ 1y 2 − √ ⎠ (ac − b 2 )ϕ 1y − bϕ 2y ϕ 2y ( ) ϕ1x ϕ = det 1y ϕ 2x ϕ 2y (√ ) 2 ( ) (ac − b2 ) ϕ2y ϕ = det 1y a −ϕ 1y ϕ 2y (√ ) 2 (ac − b2 ) (ϕ ) 2 = a 2y + ϕ 2 1y ≠0.
6.3. CHARACTERISTICS AND HIGHER ORDER EQUATIONS 43 Thus, by our construction of ϕ, wehave which implies aϕ 2 x +2bϕ x ϕ y + cϕ 2 y =0 a (ϕ 1x + iϕ 2x ) 2 +2b (ϕ 1x + iϕ 2x )(ϕ 1y + iϕ 2y )+c (ϕ 1y + iϕ 2y ) 2 =0. Collecting the real parts we have a ( ϕ 2 1x − ϕ 2 2x) +2b (ϕ1x ϕ 1y − ϕ 2x ϕ 2y )+c ( ϕ 2 1y − ϕ 2 2y) =0. We now collect the terms involving ξ = ϕ 1 on the left and those involving η = ϕ 2 on the right to get aξ 2 x +2bξ x ξ y + cξ 2 y = aη 2 x +2bη x η y + cη 2 y which from (6.3.10) gives ã = ˜c. Next we collect the imaginary parts to get 2 (aϕ 1x ϕ 2x + b ( ) ) ϕ 2x ϕ 1y + ϕ 2y ϕ 1x + cϕ1y ϕ 2y =0, which implies ˜b = aξx η x + b ( ξ y η x + ξ x η y ) + cξy η y =0. Also, since ˜b = 0, we know from invariance that −ã˜c = ˜b 2 − ã˜c 0, y > 0, into canonical form. In this case we have dy dx = iy x = b + i√ ac − b 2 a
Page 1 and 2: Chapter 6 Partial Differential Equa
Page 3 and 4: 6.1. INTRODUCTION 3 In this case we
Page 5 and 6: 6.1. INTRODUCTION 5 Solutions to La
Page 7 and 8: 6.1. INTRODUCTION 7 t u=0 u=1/2 u=1
Page 9 and 10: 6.2. LINEAR AND QUASILINEAR EQUATIO
Page 31 and 32: 6.3. CHARACTERISTICS AND HIGHER ORD
Page 41: 6.3. CHARACTERISTICS AND HIGHER ORD
Page 53 and 54: Bibliography [1] R. Dautray and J.L

Chapter 6 Partial Differential Equations

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