模块和表示论代写 MATH 5735代写 数学作业代写 数学代写
616MATH 5735 - Modules and Representation Theory Assignment 1 模块和表示论代写 1. (9 marks) Recall that an integral domain is a commutative ring (with unity) that has no zero divisors. (a) Pro...
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数学偏微分方程代写 All explanations and calculations must be shown in full to receive full credit. You may use the book but no electronic computation devices.
All explanations and calculations must be shown in full to receive full credit.
You may use the book but no electronic computation devices.
This paper contains 4 questions that carry a total of 100 points.
utt(x, t) = uxx (x, t) + q(x, t), −∞ < x < ∞, t > 0,
u(x, t), ux (x, t) → 0 as x → ±∞, t > 0,
u(x, 0) = 0, ut(x, 0) = 0, −∞ < x < ∞,
sketch the domain of integration generated by the above representation formula in the upper half of the (ξ, τ )-plane and compute u(−1, 4). (25 points)
uxx (x, y) + 6uxy(x, y) + 9uyy (x, y) − 3ux (x, y) − 9uy (x, y) − 108u(x, y) = 18(5 + 18x − 18y), − ∞ < x < ∞, y > 0.
(i) Determine the type of the equation and perform a suitable transformation of coordinates to bring the equation to its canonical form.
(ii) Find the general solution of the canonical form of the equation and, hence, the general solution of the given PDE.
ut(x, t) + 4tux(x, t) = 3x − 6t2 − u(x, t), −∞ < x < ∞, t > 0,
u(x, 0) = 4 − 2x, −∞ < x < ∞.
Sketch the correct shape of the characteristics. (25 points)
uxx(x, y) + (2 + ε)ux(x, y) − εuy(x, y) − (3 + ε)u(x, y) = 3x − 2 + ε(x − 1 − 2ex ), 0 < x < 1, −∞ < y < ∞,
u(0, y) = 2y + 4ε, u(1, y) = 2ey − 1 + 4εe−3 , −∞ < y < ∞, 0 < ε << 1.
Use the method of asymptotic expansion to compute the first two nonzero terms of an approximate solution of this BVP. (25 points)
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