数学偏微分方程代写 4143/6543代写 偏微分方程代写代写

4143/6543 Partial Differential Equations

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.

1. Consider the Cauchy problem

u_tt(x, t) = u_xx (x, t) + q(x, t), −∞ < x < ∞, t > 0,

u(x, t), u_x (x, t) → 0 as x → ±∞, t > 0,

u(x, 0) = 0, u_t(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)

2. Consider the partial differential equation 数学偏微分方程代写

u_xx (x, y) + 6u_xy(x, y) + 9u_yy (x, y) − 3u_x (x, y) − 9u_y (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.

3. Use the method of characteristics to solve the initial value problem 数学偏微分方程代写

u_t(x, t) + 4tu_x(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)

4. Consider the boundary value problem 数学偏微分方程代写

u_xx(x, y) + (2 + ε)u_x(x, y) − εu_y(x, y) − (3 + ε)u(x, y) = 3x − 2 + ε(x − 1 − 2e^x ), 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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