Question: 1. Determine the adjoint operator L* and corresponding adjoint boundary condi- tions for the following boundary value problem: Lu = u(x) - (sin x) u'

1. Determine the adjoint operator L* and corresponding adjoint boundary condi- tions for the following boundary value problem: Lu = u"(x) - (sin x) u' (x) + 3u(x) = 0, 05x51 u(0) = 0 u'(0) = 0 u"(0) = 4u( T) The symbol ' represents derivative with respect to the independent variable x.som : 1 operator is defined as ( ) (1) x) for Lu = ullens - (sinx)ulus+ 3 4(x)=0, 0EXSIT lal . B. e. 4 0 ) = 0, 1/ 40-0-0 ul ( 0 ) = 0 41 20 ) = 441D Thenn IT = (u"(*) - sinxu + sw. vax ibb smells = [ ullv dx- / u( sinx v> dx + 3fundx Using toomula integration by ports- O = ( 17 ) V Stag = fg- SadF (OJVA = ( 71 )/ V IT = [u"v]. - [ u"v'dx+ ud( sinxv) -[usin x]." + 3 / uvdy Juvax + Ju ( cosx y + sinx v ' + 3 v ) d x O There fore u satisfies boundry faluc Conditions- U ( 0 ) = 0, U 10) = 0, ull10) = QUITD 20 , = U' ( TDV( TT ) - U" LOSVIOS - UITDV ITTY - ULOV ' 10) + UCTT)~" (TD) - uCODY (of + Su[- v"+ (cosx+ 3 )y +sinxv ' ] dx V 10 ) - U/ TT ) V / ( IT ) + O( TT ) V " ITT Su [- ul + sinxN't (cosx+ 3) vax]it v ( x ) = 0 , V ( TT ) = 0 , V " ( TD ) = 4v (0 ) then LLY,NY = 0-0-uto+Jur-d2 dxz + sinxd. dx + ( cos* +3) vax ] LLY,V>=

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