Lesson 41. where the linear operator x xy y. to enforce the one to one correspondence of this transformation, we assume

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1 Modle 4: Partial Differential Eqations Lesson 4 Classification of emi-linear nd order Partial Differential Eqations 4. Classification of nd order Partial Differential Eqations: Parabolic Hyperbolic Elliptic Eqations A nd order semi linear partial differential eqation can be pt in the form L + g(, y,,, ) = 0 (4.) y where the linear operator L Ry (, ) + y (, ) + Ty (, ) y y is sch that the coefficient fnctions R, and T are continos fnction of, yand R + + T 0. We change the independent variables ( yto, ) (ξ,η ) as ξ = ξ( y, ) and η = η( y, ), to enforce the one to one correspondence of this transformation, we assme ξη ηξ 0. y y The coefficients and the partial derivatives in the given eqation are written in terms of the transformed variables. The first and second order partial derivatives become: = ξξ + η; y = ξξy + ηy = ξξ + ηξ + ξ + ξη + ηη + η y ξξ y ξη y ξ y ξη y ηη y η y = ξ + ηξ + ξ + ξη + η + η ξξ ξη ξ ξη ηη η

2 Classification of emi-linear nd order Partial Differential Eqations = ξ + ηξ + ξ + ξη + η + η yy ξξ y ξη y y ξ yy ξη y y ηη y η yy R + + T = ξξ Rξ + ξξ + Tξ y yy ( y y ) + ξη ( Rηξ + ( ηξ + ξη ) + Tξη ) y y y y + R + + T + F. ηη ( η ηη y ξy ) ( ξη,, ξ, η, ) Eqation (4.) becomes A( ξ, ξ ) + B( ξ, ξ ; η, η ) + A( η, η ) = G( ξ, η,,, ) (4.) y ξξ y y ξη y ηη ξ η where A(, v) = R + v + Tv (4.3) B(, v;, v) = R + ( v + v) + Tvv (4.4) Now, the problem is to determine ξ &η so that the eqation (4.) takes the simplest possible (Canonical) form. When the sign of the determinant of the qadratic form (4.3) is everywhere positive, negative or zero it is easy to make the classification. Case A: When > 0 everywhere in the domain. The new independent variable ξ &η can be so chosen that the coefficients of ξξ and ηη in (4.) vanish. The roots λ & λ of the eqation Rα α T + + = 0 are real and distinct. The coefficient of ξξ & ηη in (4.) will vanish if we close ξ &η sch that

3 Classification of emi-linear nd order Partial Differential Eqations ξ ξ y y + λ ; = λ y y A sitable choice will be ξ = f( y, ), η = f( y, ) where f( y, ) = c, f( y, ) = c are the soltion of the ordinary differential eqations d + λ( y, ) = 0; + λ( y, ) = 0respectively. d It can be verified that A ξ ξ A η η B ξ ξ η η = RT ξ η ξ η (4.5) (, y) (, y) (, y;, y) (4 )( y y ) /4. Now when the A s are zero; B = ( 4 RT )( ξη ξη ) y y ince > 0 B > 0, hence eqation (4.) redces to = φξη (,, ξ, η, ) (4.6) ξη The crves ξ ( y, ) = constant, η ( y, ) crves of eqation (4.). Eqation (4.6) is called the canonical form of eqation = constant are called the characteristic Eample Redce the eqation = 0 to a canonical form. yy

4 Classification of emi-linear nd order Partial Differential Eqations oltion comparing with the standard form, we note that R=, = 0, T = Then RT = 4 > 0. o + + = 0 becomes α = 0 α =±. Rα α T λ = ; λ = Now d + = + = 0 y c d = = 0 y c Taking ξ = y+ ; η = y = ξ + η = + ( ) = ξ η ξ η ξ η y = ξ + η, = ξξ ξη + ηη + ξ η yy = 0becomes ξη ξ η, = + + yy ξξ ξη ηη = ( ) ( ). 4 = ξ η 4 ξ η ( ) Case B: If = 0 Roots of the eqation Rα α T + + = 0 are real and eqal. We define ξ as in case A and take η to be any fnction of y, which is independent of ξ. In this case we have A( ξ, ξ ) = 0 as before and hence from eqation (4.5), B = 0. y Bt A( η, η ) 0 ξ & η are independent fnctions. y

5 Classification of emi-linear nd order Partial Differential Eqations Hence the canonical form in this case is = φξη (,,,, ) y (4.7) η Eample + + = 0canonical form. y yy oltion comparing with the standard form, we note that R=, =, T =, and = 0. ( ) Rα + α + T = α + α + = α + = 0 α =,. = 0 y= c, take ξ = y d Then chose η = + y. Using these ξ &η : we have the canonical form as ξ = ηf( ξ) + f( ξ) where ξ η = 0 f & f are arbitrary fnctions. Hence the soltion of the given eqation is: z = ( + y) f ( y) + f ( y). Case C: < 0. In this case, the roots of the eqation Rα α T + + = 0 are compleconjgates. Proceeding as in case A; the canonical form = φξη (,,,, ) y. η

6 Classification of emi-linear nd order Partial Differential Eqations Bt ξ &η are comple conjgates.to get the real canonical form, we se the transformation α = ( ), ( ) ξ + η β = i η ξ = +. ξ η 4 α β o the canonical form in this case is + = ϕαβ (,,,, ) α β. α β Eample 3 Redce the eqation + = 0 to canonical form. yy oltion Clearly, R T =, = 0, =, and < 0. α + = 0 α =± i, henceλ = i; λ = i, ξ = iy + ; η = iy +, ; α = β = y αα + ββ = α is the canonical form α No we classify second order eqation of the type (4.) by their canonical form as: A) Hyperbolic if > 0, B) Parabolic if = 0, C) Elliptic if < 0. Clearly the one dimensional wave eqation given by the Hyperbolic eqation, tt = c is an eample for

7 Classification of emi-linear nd order Partial Differential Eqations the one dimensional heat condction eqation given by t for the parabolic eqation and the Laplace eqation the elliptic eqation. = α is an eample + yy = 0 is an eample for Eample 4 Discss the natre of the eqation + + y = y ( ) yy 0 oltion Clearly RT = ( + y ). Hence the given eqation ishyperbolic at all points ( y, ) sch that + y >, Parabolic if + y = and Elliptic if + y <. Eercises. Redce the eqation to its canonical form and classify it: tt + 4t t = 0.. Classify the partial differential eqation: tt + (5 + ) t + ( + )(4 + ) = 0. Keywords: Elliptic,Hyperbolic, Parabolic, References Ian neddon, (957). Elements of Partial Differential Eqations. McGraw-Hill, ingapore

8 Classification of emi-linear nd order Partial Differential Eqations Amaranath.T, (003). An Elementary Corse in Partial Differential Eqations.Narosa Pblishing Hose, New Delhi ggested Reading I.P. tavrolakis, tephen a tersian, (003). Partial Differential Eqations. Allied Pblishers Pvt. Limited, New Delhi J. David Logan, (004). Partial Differential Eqations. pringer(india) Private Ltd. New Delhi