Mathematics-IV Unit 2 Notes AKTU (BAS303)

AKTU · BTECH · Semester 3, 4 · Mathematics-IV · Unit 2 · Notes

AKTU Mathematics-IV (BAS303) Unit 2 notes for B.Tech Semester 3/4 – Applications of PDE and Fourier Transform. Topics: Method of Separation of Variables…

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Unit 2: Applications of PDE and Fourier Transform – AKTU syllabus topics

  • Method of Separation of Variables
  • Heat and Wave Equations: One Dimensional Heat Equation, One Dimensional Wave Equation, Two Dimensional Laplace Equation
  • Fourier Transform and Inverse: Complex Fourier Transform, Fourier Sine Cosine Transform, Convolution Theorem, Fourier Transform PDE Application

Most asked AKTU PYQ questions from Unit 2

  1. Q4b. Solve the equation ∂²u/∂x² + ∂²u/∂y² = 0 subject to the boundary conditions, 𝑢(0,𝑦) = 𝑢(𝑙,𝑦) = 𝑢(𝑥,0) = 0 and 𝑢(𝑥,𝑎) = sin(nπx/l) (10 marks, 2020-21)
  2. Q2b. Calculate the deflection u(x,t) of a tightly stretched vibrating string of unit length that is initially at rest and whose initial position is given by u(x,0) = 𝑠𝑖𝑛𝜋𝑥 + (1/3)𝑠𝑖𝑛3𝜋𝑥 + (1/5)𝑠𝑖𝑛5𝜋𝑥, 0 < 𝑥 < 1 (10 marks, 2020-21)
  3. Q4a. A rod of length 𝑙 with insulated sides is initially at a uniform temperature 𝑢₀. Its ends are suddenly cooled to 0°C and are kept at that temperature. Calculate the temperature function 𝑢(𝑥, 𝑡). (10 marks, 2020-21)
  4. Q4b. A string is stretched and fastened to two points l apart, motion is started by displacing the string into the form y = k(lx – x²) from which it is released at time t=0. Find the displacement of any point on the string at a distance of x from one end at time t. (7 marks, 2024-25)
  5. Q1c. Classify the following Partial Differential Equation 4∂²u/∂x² - 4∂²u/∂x∂y = 0 (2 marks, 2020-21)
  6. Q2b. A string is stretched and fastened to two points l meter apart. Motion is started by displacing the string in the form u(x,0) = A sin(πx/l) from which it is released at time t=0. Show that the displacement of any point at a distance x from one end at time t is given by u(x,t) = A sin(πx/l)… (10 marks, 2022-23)
  7. Q4b. Solve the Laplace equation u_xx + u_yy = 0, x∈(0,1), y∈(0,1) with conditions u(x,0) = u(x,1) = 0, u(0,y) = 0, u(1,y) = f(y) by method of separation of variables. (10 marks, 2022-23)
  8. Q4a. Solve by the method of separation of variables, the heat equation u_t = u_xx, 0 < x < 1, t > 0 subject to conditions u(x,0) = x - x², u(0,t) = u(1,t) = 0. (10 marks, 2022-23)
  9. Q2b. Determine the solution of one dimensional heat equation ∂u/∂t = ∂²u/∂x² where the boundary conditions are u(0,t) = 0, u(l,t) = 0 (t > 0) and the initial condition u(x,0) = 3sin(πx/l), l being the length of the bar. (10 marks, 2021-22)
  10. Q4b. Determine the solution of Laplace equation ∂²u/∂x² + ∂²u/∂y² = 0 subject to the boundary conditions u(0,y) = u(l,y) = u(x,0) = 0 and u(x,a) = f(x). (10 marks, 2021-22)

AKTU paper codes: BAS303, BAS303H, BAS403, BAS403H, KAS302, KAS402

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