Can anyone help with this question please?
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Can anyone help with this question please? Given a domain Ω c R2 and a smooth function f,uo : Ω-+ R consider the proble...
Can anyone help with this question please? Many thanks!!!!! Let Ω Rn be a bounded domain and f : Ω-, R and g : 0Ω-+ R b...
Can anyone help with this question please? Many thanks!!!!!
Let Ω Rn be a bounded domain and f : Ω-, R and g : 0Ω-+ R be given functions. Consider the PDE problem -Au = f in Ω, where n is the external unit normal of Q. Show that there is at most one solution u E C2(Q) n Co (O). For this purpose, use an energy argument as before but amend the energy as appropriate.
Let Ω Rn be...
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) ...
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the Fourier transformed version of (2). Then, find the solution of this transformed version u(t,)-((,) (b) Invert the solution in part (a) to get the solution, u(t, x)-F-(u)(t, x), to (2)
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
can please explain why F(sigma)= u? We consider the PDE: for given o(t) € H|(12) find...
can please explain why F(sigma)= u?
We consider the PDE: for given o(t) € H|(12) find the (weak) solution u € H}(2) of V. (g(x)Vu(x)) = 1. The corresponding parameter-to-solution map is defined as F: D(F):= H (1) C H²(2) L’(1) F(o)= u uc H (12) c L’(2) solving b(u, w;o) = f(w) for all w e H7(), b(u, w; 0) := ( D2.Vwdi, f(w):=- / w dr. The associated inverse problem is for u E L(12) find o E...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Question For this problem, consider the function on the domain of all real numbers. (a) The value of is Oo . (If you ne...
Question For this problem, consider the function on the domain of all real numbers. (a) The value of is Oo . (If you need to use -co or o, enter-infinity or infinity.) (b) The value of f(x) limx→- is O0 X . (If you need to use -oo or co, enter -infinity or infinity.) (c) There are two x-intercepts; list these in increasing order: s- t- (d) The intercepts in part (c) divide the number line into three intervals. From...
Problem C The partition function for an ideal gas is given by integrating over all possible...
Problem C The partition function for an ideal gas is given by integrating over all possible position and momen- tum configurations, weighted by a Boltzmann factor, for each particle (6 integrals per particle over z, v, z, pz, py, pz _ each running from-oo to +oo) and multiplying all N of these together (the factor of h is included to cancel the dimensions of dpdr; the factor of N! is included to divide out the multiplicity of particle-particle exchange) a)...
Can anyone help with this question please? Many thanks!!!!!
Let Ω Rn be a bounded domain and f : Ω-, R and g : 0Ω-+ R be given functions. Consider the PDE problem -Au = f in Ω, where n is the external unit normal of Q. Show that there is at most one solution u E C2(Q) n Co (O). For this purpose, use an energy argument as before but amend the energy as appropriate.
Let Ω Rn be...
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the Fourier transformed version of (2). Then, find the solution of this transformed version u(t,)-((,) (b) Invert the solution in part (a) to get the solution, u(t, x)-F-(u)(t, x), to (2)
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
can please explain why F(sigma)= u?
We consider the PDE: for given o(t) € H|(12) find the (weak) solution u € H}(2) of V. (g(x)Vu(x)) = 1. The corresponding parameter-to-solution map is defined as F: D(F):= H (1) C H²(2) L’(1) F(o)= u uc H (12) c L’(2) solving b(u, w;o) = f(w) for all w e H7(), b(u, w; 0) := ( D2.Vwdi, f(w):=- / w dr. The associated inverse problem is for u E L(12) find o E...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Question For this problem, consider the function on the domain of all real numbers. (a) The value of is Oo . (If you need to use -co or o, enter-infinity or infinity.) (b) The value of f(x) limx→- is O0 X . (If you need to use -oo or co, enter -infinity or infinity.) (c) There are two x-intercepts; list these in increasing order: s- t- (d) The intercepts in part (c) divide the number line into three intervals. From...
Problem C The partition function for an ideal gas is given by integrating over all possible position and momen- tum configurations, weighted by a Boltzmann factor, for each particle (6 integrals per particle over z, v, z, pz, py, pz _ each running from-oo to +oo) and multiplying all N of these together (the factor of h is included to cancel the dimensions of dpdr; the factor of N! is included to divide out the multiplicity of particle-particle exchange) a)...
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