偏微分代写 Partial differential代写

Partial differential

偏微分代写 Prob1(Total of 25 pts) 1a)(15pts)Construct the Fourier series representation of the solution, calculating any integrals explicitly.

Prob1(Total of 25 pts)

Consider the Dirichlet IBVP for 0<x <π, with homoge- neous boundary data:

u_t=μu_xx -vu_xxxx ,t>0,0<x<π,μ>0,v>0,

u(x,t=0)=δ(x- π/2),

u(x=0,t)=u(x=π,t)=0, (0.1)

where δ (x – π/2) denotes the “Dirac delta function”.

1a)(15pts)Construct the Fourier series representation of the solution, calculating any integrals explicitly.

1b)(10pts)What function of(x,t) does the solution look like for large time t? That is, for large t, what is an expression for the dominant behavior of u(x,t)?

 

 

Prob 2 (Total of 25 pts)  偏微分代写

Consider the(IBVP) for wave equation on the line:

u_tt − c²u_xx = 0, t>0, −∞<x<+∞, c>0,

u(x,t= 0) =u_in(x),

u_t(x,t= 0) =v_in(x),

u(x,t) → 0_asx → ±∞.

2.a (10 pts) Let u^(a)(x,t) denote the solution of the IBVP for the following initial

data:

u^(a)(x,t= 0) =δ(x),

u_t^(a)(x,t= 0) = 0,

where δ(x) denotes the ”Dirac delta function”. Using d’Alembert’s solution formula,

write down the solution to this initial value problem. For fifixed t >0, sketch the solution u^(a)(x,t) as a function of x. Use ”arrows” to indicate the direction of the motion of the wave.

2.b (10 pts) Let u^(b)(x,t) denote the solution of the IBVP for the following initial

data:

u^(b)(x,t= 0) = 0,

u_t^(b)(x,t= 0) =δ(x),

where δ(x) denotes the ”Dirac delta function”. Using d’Alembert’s solution formula, write down the solution to this initial value problem, doing the integral analytically. For fifixed t >0, sketch the solution u^(b)(x,t) as a function of x. Use ”arrows” to indicate the direction of the motion of the wave.

2.c (5 pts) Let u^(c)(x,t) denote the solution of the IBVP for the following initial data:

u^(c)(x,t= 0) =δ(x),

u_t^(c)(x,t= 0) =δ(x).

Using the linearity of the wave equation, represent the solution u^(c) in terms of u^(a) and u^(b) . For fifixed t >0, sketch the solution u^(c).

 

 

 

 

3 (Total of 25 pts)  偏微分代写

Consider the dispersive wave equation

u_t+cu_x +u_xxx = 0, t>0, −∞<x<+∞ , c>0,

u(x,t= 0) =u_in(x).

Prob 3a (9 pts): Construct (i) the plane wave solution u(x,t;k) =exp[i(kx−w(k)t)],

and explicitly state it’s (i) dispersion relation, (ii) spatial frequency, (iii) temporal frequency, (iv) phase velocity, and (v) group velocity.

Prob 3b (3 pts): Give the Fourier formulas for the unique solution of the IVP, as a superposition of plane waves.

Prob 3c (3 pts): State the formula (derived in the class notes) for the long time behavior of this solution,

u(x,t) =??,ast→+∞, x→+∞, x/t=v_gp.

Be sure that you define explicitly all terms in this formula.

Prob 3d (10 pts): Then describe in words the qualitative behavior of the solution u(x,t) for large t. That is, describe in words ”physical behavior” of the wave as represented by the formula. In your description, use terms such as ”plane wave”, ”spatial frequency”, ”temporal frequency”, ”phase velocity”, ”group velocity”, and ”temporal decay of the wave amplitude”. Conclude your description with an illustration the ”prism e↵ect”.

 

 

4 (Total of 25 pts)  偏微分代写

Consider the IVP for the nonlinear transport equation on the line:

u_t +u²u_x = 0, t>0, −∞<x<+∞,

u(x,t= 0) = +2, x<0;

u(x,t= 0) =−1, x>0.

Prob 4a (15 pts): Solve this equation by the method of characteristics: i) Give the IVP for the ODE that defifines the characteristics X(t;X₀); ii) Solve this ODE for the family of characteristics indexed by X0; iii) In terms of this family of characteristics, give the formula for the solution u; iv) Sketch the characteristics on the x−t plane, showing the regions of the x-t where the value of the solution u is known, and showing the regions where the solution u is not defifined by the method of characteristics; v) Where the solution is defifined by the method of characteristics, show the value of the solution on the x−t plane.

Prob 4b (10 pts): Using the notion of a ”weak solution”, extend your solution to the entire upper half x −t plane, with no ”omitted regions”. This will require fifitting a shock through the interior of the excluded ”wedge”. Give the shock speed, and the value of the solution u to the left and the right of the shock. Extend your sketch in problem 4a of the solution by the method of characteristics, to the entire upper half x−t plane, with no omitted regions.

 

 

 

 

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