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Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x,...
au du atua = 90, дх with the initial conditions at t = 0: u=0 if...
au du atua = 90, дх with the initial conditions at t = 0: u=0 if u=-1-1 u=1 - << -1, if -1 <I<0, if 0 < I< 0. (Define u(r, t), x,t and the constant qo appropriately.) (b) Use the method of characteristics along suitable curves r(t) to obtain the implicit equation satisfied by the general solution ur,t) of the PDE given in the first problem (do not have to use the initial conditions at this stage, so there...
Solve the wave equation on the domain 0 < x < , t > 0 ?...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
Solve the equation for u(x, t) if it satisfies the equation: with boundary and initial conditions...
Solve the equation for u(x, t) if it satisfies the equation: with boundary and initial conditions given by where δ(x) and δ(t) are Dirac delta functions. 2. du-Ka_ = δ(x-a)s(t) for 0 < x < oo; t > 0 at ах? du ах (0, t) = 0;u(co, t) =0;(mt) = 0; u(x, 0)=0 ox
Solve the equation for u(x, t) if it satisfies the equation: with boundary and initial conditions given by where δ(x) and δ(t) are Dirac delta functions....
-). Solve the initial and boundary value problem: uUx=0, TE (0,), t > 0, U (0,t)...
-). Solve the initial and boundary value problem: uUx=0, TE (0,), t > 0, U (0,t) = u(,t) = 0, >0, u(,0) - cos', 1€ (0,7).
** Lon u. that the solution to the heat conduction problem aug , 0<<L t >...
** Lon u. that the solution to the heat conduction problem aug , 0<<L t > 0 u(0,t) - 0, u(L,t) = 0 (u(a,0) = f (3) is given by u(3,4) – È che+n*/2°' sin (182), – Ž Š 5(2) sin (%), vnen. Explicitly show by substitution that this function u(x, t) satisfies the equation aus = U, and all of the given boundary conditions. Note: You can interchange/swap sums and derivatives for this function (that doesn't always work!).
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
3) Using the Method of Variation of Parameter, solve the following linear differential equation y' (1/t)...
3) Using the Method of Variation of Parameter, solve the following linear differential equation y' (1/t) y 3cos (2t), t > 0, and show that y (t) 2 for large t
5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0...
5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0 1 <1 > 1 u(t,0) = Then draw the solution for different values of time.
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the...
Find the solution to the heat equation on the infinite
domain
∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1.
in terms of the error function.
Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T...
+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T Graph the solution (you may use Excel or Matlab) for T= 1sec, 0.1sec, 0.01sec, and 0.001sec. Do you see what is happening to the output? What is happening to the input?!
au du atua = 90, дх with the initial conditions at t = 0: u=0 if u=-1-1 u=1 - << -1, if -1 <I<0, if 0 < I< 0. (Define u(r, t), x,t and the constant qo appropriately.) (b) Use the method of characteristics along suitable curves r(t) to obtain the implicit equation satisfied by the general solution ur,t) of the PDE given in the first problem (do not have to use the initial conditions at this stage, so there...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
Solve the equation for u(x, t) if it satisfies the equation: with boundary and initial conditions given by where δ(x) and δ(t) are Dirac delta functions. 2. du-Ka_ = δ(x-a)s(t) for 0 < x < oo; t > 0 at ах? du ах (0, t) = 0;u(co, t) =0;(mt) = 0; u(x, 0)=0 ox
Solve the equation for u(x, t) if it satisfies the equation: with boundary and initial conditions given by where δ(x) and δ(t) are Dirac delta functions....
-). Solve the initial and boundary value problem: uUx=0, TE (0,), t > 0, U (0,t) = u(,t) = 0, >0, u(,0) - cos', 1€ (0,7).
** Lon u. that the solution to the heat conduction problem aug , 0<<L t > 0 u(0,t) - 0, u(L,t) = 0 (u(a,0) = f (3) is given by u(3,4) – È che+n*/2°' sin (182), – Ž Š 5(2) sin (%), vnen. Explicitly show by substitution that this function u(x, t) satisfies the equation aus = U, and all of the given boundary conditions. Note: You can interchange/swap sums and derivatives for this function (that doesn't always work!).
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
3) Using the Method of Variation of Parameter, solve the following linear differential equation y' (1/t) y 3cos (2t), t > 0, and show that y (t) 2 for large t
5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0 1 <1 > 1 u(t,0) = Then draw the solution for different values of time.
Find the solution to the heat equation on the infinite
domain
∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1.
in terms of the error function.
Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T Graph the solution (you may use Excel or Matlab) for T= 1sec, 0.1sec, 0.01sec, and 0.001sec. Do you see what is happening to the output? What is happening to the input?!
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