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1. TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS UNIT-I PART -A 1. Write down the Dirichlets conditions for a function to be expanded as a Fourier series. 2. What is the constant term a0 and the co.efficient of cosnx ,an in the fourier series expantion of f(x)=x-x3 3. State parsevals identity for full range expansion of f(x) as fourier series in (0,2 l) 4. Find an in expanding e-x as a fourier series in(-π, π) 5. In the fourier expansion of in (-π, π). 6. If f(x)=2x in the interval (0,4) then f
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1. TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS UNIT-I PART -A1.   Write down the Dirichlets conditions for a function to be expanded as a Fourier series.2.   What is the constant term a 0 and the co.efficient of cosnx ,a n in the fourier series expantionof f(x)=x-x 3  3.   State parsevals identity for full range expansion of f(x) as fourier series in ( 0,2 l)  4.   Find a n in expanding e -x as a fourier series in(-π, π)5.   In the fourier expansion of                  in (-π, π).6.   If f(x)=2x in the interval (0,4) then find the value of a 2 in the fourier series expansion.7.   Find the sine series for the function f(x)=1; 0<x<π.8.   If f(x) is discontinuous at x=a,What does its fourier series represents at that point.9.   Determine the value of a n in the fourier series expansion of f(x)=x 3 in b –π <x < π.10.   If f(x)=sinhx is defined in -π<x< π, write the value of a 0 a n.  PART-B1.   If f(x)=  (π-x) find the fierier series of period 2π in the interval (0,2π),Hence deduce that 1- -  +  -  +……… =  .2.   If              , find a Fourier series of periodicity and 2π and hence evaluate  +  +  +………..3.   Find the fourier series of f(x)=xsinx in –      .4.   Obtain cosine series for f(x)=x sinx in 0     .5.   Find the half range cosine series for the function f(x)=x in  and deduce the sum of series    +   +    +……………6.   Obtain the half range cosine series f(x)=(x-2) 2, 0≤x≤2 and deduce that     =    .7.   Find the cosine series for πx -   in 0 < x <  and deduce     =     .8.   Find the fourier series of the function               .9.   Compute the first three harmonic of the fourier series of the given by the following table.X 0 π/3 2π/3 π 4π/3 5π/3 2πF(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0    10.   Find the fourier series for y=f(x) upto second harmonic form the following datax 0 1 2 3 4 5y 9 18 24 28 26 20 UNIT-2 2MARKS1.   Write the Fourier transform pair.2.   Prove that Fc   =      .  3.   Find the Fourier cosine transform of             .4.   Find fourier sine transform of F(x)=   (a>o).5.   Find the fourier transform of f(x) defined by         .6.   State parseval’s identity theorem in Fourier transform.7.   Find the Fourier transform of f(x)=      .8.   Find the fourier transform of f(x) as                  .9.   State convolution theorem on fourier transform.10.   Find the Fourier sine Transform of f(x)=1 in (0,1).8-MARKS1.   Find the Fourier transform of the function f(x)defined by         .Hence provethat(i)        ds =   (ii)      ds=  .2.   Find the Fourier transform of       .Hence deduce that     4 dt=  .3.   Find the fourier sine transform of x   .4.   Find the Fourier inverse sine cosine transform of    ,a>0.5.   Evaluate           using Fourier cosine transform.6.   Using parsevel’s identity ,evaluate (i)         dx,a>0(ii)       a>0.7.   Find the Fourier cosine transform of    .Hence deduce the value of        .8.   Find Fourier sine and transform of    and hence find the Fourier sine transform of    andFourier cosine Transform of    .    UNIT 3 2MARKS1.   Form the P.D.E form z=               .2.   Form the P.D.E by eliminating the constants a and b from z=      .3.   Solve   +   =1.4.   Solve     +     =   .5.   Solve z=   +   .6.   Find the complete solution and singular solution of z=px + qy+   -    7.   Find the singular solution ofz=px+qy+        .8.   Solve               9.   Solve               10.   Solve             8MARKS1.   Solve                 2.   Solve                   .3.   Solve                 .4.   Solve∂ 2 z/∂x 2 + ∂ 2 z/∂x∂y -2∂ 2 z/∂y 2 = sinh(x+y) + xy.5.   Solve              .6.   Solve ∂ 3 z/∂x 3 -2  3 z/∂x 2 ∂y =       .7.   Solve               .8.   Solve                 .9.   Solve                UNIT-4 Applications of Partial Differential equations2 Marks1)   Classify (1+x) 2 u xx -4u xy + u yy [if x=1, Δ=0 (Ans: 1.the p.d.e is parabolic), 2. if x= + or -, Δ>0(thep.d.e is hyperbolic).2)   Classify б 2 u/бxбy = (бu/бx) + (бu/бy) +xy (Ans: The p.d.e is hyperbolic).3)   What conditions are assumed in deriving the dimensional wave equation.4)   In the wave equation б 2 y/ бt 2 = c 2 (б 2 y/бx 2 ). What is c 2 stands for? (or) State the physicalmeaning for the constant in ODWE.[Ans:c 2 =T/m=tension/Mass per unit length]    5)   A string is stretched and fastened to two point l apart. Motion is started by displacing thestring into form y(x, 0) = y 0 sin (πx/l) from which it is released from rest at time t=0 formulatethe B.V.P.6)   Give the various solutions of wave solutions.7)   Classify u xx + u yy = (u x ) 2 + (u y ) 2 (Ans: P.D.E is elliptic)8)   Classify the p.d.e u xx + 4u xy + 4u xy - 12u x + u y + 7u = x 2 + y 2 (Ans: P.D.E is hyperbolic).9)   Classify the p.d.e u xx + xu yy = 0. (Ans: 1. if x<0, p.d.e is elliptic. 2. If x=0, p.d.e is parabolic. 3. If x>0, p.d.e is hyperbolic)10)   Classify б 2 u/бx 2 + 2б 2 u/бxбy + б 2 u/бy 2 + бu/бx – бu/бy =0.(Ans: p.d.e is hyperbolic)8 Marks1)   Derive the solution for the one-dimensional wave equation by the method of separationvariable.[Ans: y(x,t) = (C 9 x + C 10 )(C 11 t + C 12 )]2)   A string is stretched and fastened to two points l apart. Motion is started by displacing thestring into the form 50(lx – x 2 ) (or) K (lx – x 2 ) from which it is released at time t=0. Find thedisplacement of any point on the string at a distance x from one end at time t.3)   A taut string of length 2l is fastened at both ends. The midpoint of the string is taken to aheight b and then released from rest in that position. Find the displacement of any point of the string at any subsequent time.4)   A tightly stretched string with fixed end points x=0 and x=1 is initially at rest in its equilibriumposition. If it is set vibrating by giving each point a velocity λx (l-x), find the displacement of the string at any distance x from one end a any time.5)   A string is stretched between two fixed points at a distance 2l apart and the points on thestring are given initial velocities V, whereV=Cx/l: 0<x<1 , find the displacement.c(2l-x)/l: l<x<2l6)   The ends A and B of a road 30cms long have their temperatures kept at 20°C and 80°C untilsteady state prevails. The temperature at the end B is then suddenly reduced to 60°C andthat of A is raised to 40°C and maintained so. Find the temperature distribution.7)   A metal bar 10cm long is insulated sides, has its ends A and B kept at 20°C and 40°Crespectively until steady state conditions prevail. The temperature at A is then suddenlyraised to 50°C and B is lowered to 10°C. Find its subsequent temperature at any point at thebar at any time.8)   A rod of length l has its ends A and B kept at 0°C and 100°C until steady state prevails. Thetemp at A is raised to 25°C and B is reduced to 75°C. Find the temp. dist.9)   A bar of 10 cm long with insulated sides has its ends A and B at temp 50°C and 100°C, untilsteady state prevails. The temp at A is suddenly raised to 100°C, B is lowered to 60°C. Findthe temp. dist.
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