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We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at อใน əx², u(0, t) = 20, u(30, t) = 50, u(x, 0) = 60 - 2x, t> 0, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).

We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at อใน əx², u(0, t) = 20, u(30, t) = 50, u(x, 0) = 60 - 2x, t> 0, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at = อน əx²¹ u(0,t) = 20, u(30, t) = 50, t> 0, u(x,0) = 60 2x, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1). d) By plotting the solution obtained in part (c) using any computer package, display the initial particle concentration, the final (or steady state) particle concentration, and the particle concentration at t = 3 and t = 25 i.e., the intermediate particle concentration. (For part (d), enclose together the coding of computer package that your group employs in your report). e) From the plot in (d), produce some plausible explanations on the intermediate particle concentration with respect to its behaviour at the boundary conditions and also the observations as t increases.
Transcribed Image Text:We all have experienced the result of someone spraying perfume in the corner of a large room. Within a fairly short amount of time, most people in the room can smell the perfume. In this case, we say that the perfume has diffused from a region of the room in which it was highly concentrated to a region of the room in which its concentration was initially low. Diffusion is the process by which the molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. This process can be investigated using a diffusion model (also known as a heat equation): ди at = อน əx²¹ u(0,t) = 20, u(30, t) = 50, t> 0, u(x,0) = 60 2x, 0 < x < 30, (1) with u = u(x, t) corresponds to the concentration of perfume particles moving over time, t, along a one-dimensional spatial domain, x. a) Calculate the steady-state concentration of perfume particles. b) Formulate the initial-boundary value problem that determines the transient distribution of perfume particles. c) Hence, using the Fourier series techniques, derive the solution u(x, t) of the PDE (1). d) By plotting the solution obtained in part (c) using any computer package, display the initial particle concentration, the final (or steady state) particle concentration, and the particle concentration at t = 3 and t = 25 i.e., the intermediate particle concentration. (For part (d), enclose together the coding of computer package that your group employs in your report). e) From the plot in (d), produce some plausible explanations on the intermediate particle concentration with respect to its behaviour at the boundary conditions and also the observations as t increases.
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