Ma2211 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
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This document contains a question paper for a civil engineering exam covering topics in transforms and partial differential equations. It has two parts with multiple choice and long answer questions. Part A contains 10 short answer questions worth 2 marks each, covering topics like Fourier series, Fourier transforms, partial differential equations, and Z-transforms. Part B contains 5 long answer questions worth 16 marks each, involving finding Fourier series expansions, solving partial differential equations, using Fourier transforms and Z-transforms to solve problems.
Ma2211 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
- 1. ., Question Paper Code:· 23768 B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2018. Third/Fifth Semester Civil Engineering MA 2211- TRANSFORMS AND PA.R'l'IAL DiFFERENTIAL EQUATIONS (Common to all branches) ·(Regulatione 2008) Time.: Three hours -Maximum: 100 marks ·Answer ALL questions. PART A- (10 x 2 = 20 marks) 1. If f(x) =x 2 in -2 < x < 2 and· f(x + 4) =f(x), then find the coefficient a0 ofits ·Fourier series. 2. Find the root mean squa're value of f(x) =cosx in (0, 2n) .. , 3. State the co:p.volution theorem for Fourier: transform. 4. Show that the Fourier transform satisfies the linearity property. 5. For1Il the p~rtia! differential equation by eliminating the arbitrary function from z=f x +y. ) • . · · .· 6. . Find the complete integral of p =2q X 0 7. Mathematically formulate the following vibratin,g string problem: "A string is _ stretched between two fixed pomts at a distance 2l apart and the points of the string are given an. initial velocity f(:x), x · being the distance from an end point". 8. . Classify the partial differential equation: uu + 2uxy + uYY =0. 9: Find the Z transform of u (n-1) . 10. State the initial valu~ theorem ofZ transform. ; BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 1
- 2. PART B - (5 >< 16 ~.. SO marks) 11. (a) (i) Find a Fourier series to represent f(x)"" x ~ x2 from.-~ ~o tt. (10) 12. (ti) "-' Obtain the constant term a0 and the first two harmonics a1 a~d a2 . in the Fourier co&ine series representation of y ""'f(x) in· (0, 6) for the given.table.oivaiues: (6) X 0 ·1 2 3 4 . 5 y 4 8 15 7 6 2 Or (b) (i) Find the half range Fourier sine· series expansion for (a) (b) L''f(x)= 2k(L~x) · {. 2kx O<x<.L/2 . ., . 1 . . Hence evaluate L ··· ····- 4 · . L12<x<L . · .. n=l (2n -1) (8) L . , (ii) Find the complex f<?rm of the Fourier series for /(x) = e-x,-l<x<l. (8) (i) Express the function f(:<) ~ {~ lxl ~ 1 as ~ Fourier integraL lxl >1 · ·· · jsin..t ·· Hence e~alu~te 0 ~ dA. . (10) (ii) Find the Fourier sine transform of ! . X (6) Or =az.t:~. .. 0 ..(i) . Find the Fourier transform of e ,a.:> and henc~ .find the Foudertransform of e~x212 ~- (8) (ii) Using Fourier · transform methods, evaluate oO • dx . J(2 .· 2R 2 b2)' a, b > 0. 0 X: +a X + .. · (8) 13. (a) (i). Solve the partial durerential equation~ z2 {p2 + q1 }z.x2 +y 2 • (8) (ii) Solve (D2 +2DD'+D'2 )ztisinh(x +y)+ e·t+2 Y.. - (8) Or (b) (i} Solve (3z-4y) p+(4x-2z)q c 2y~3x. (8) (ii) Obtain the CQmplete . solution z ""' p X + qy + p2 q2 . and singular solution of (8) 2 23768 BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 2
- 3. 14. 15. (a) A rod; · 30 em long, has its ends A and B kept at 20°C ·and 80°C _respectively until steady state conditions prevail. The temperature at e~ch end is then suddenly reduced to O"C .and kept so. Find the resulting temperature function u(x, t) taking ~ =0 at A (16) Or (b) An infinitely long plane uniform plate is bounded by two parallel edges x = 0 and· x !!; l and an edge at right angles to them. The breadth of this edge y == 0 is l and is ·maintabted at a temperatm:e f(x). All the other three edges are· at· ,;ero temperatur~. Find.the steady state temperature at any interior point of the plate. . .(16) (a) (b) (r) Using Z-transf'orm method, solve the x(n + 1)-2x(n.)e 1'given x(O) ""0. difference equation:· (10) (ii) Find the Z transform of t" and hence deduce the result: z(tk)==~T~ ddz {z v~e-i)}, where_ T is the sampling period_ with~ t == nT, n t:o;i.2,... . (6) Or (i) · Find the inverse Z trapefor'm of (z~~)(z-~J using partial fractiofis method. (8) {ii) z.2 Find the inverse ~ t!'aneform -ttsing convolution theorem. (z~af (8) .. ·.. 3 23768 BIBIN.C / ASSOCIATE PROFESSOR / MECHANICAL ENGINEERING / RMK COLLEGE OF ENGINEERING AND TECHNOLOGY 3