![Question 1) (3 Points) Consider the strong form below. Derive the weak form equation. The space used is W H-1,1]. Define the spaces Wp and Wo Dirichlet Strong Form: Find u e W2,00 such that u(x) _ 2 sin(x)a(z) u(-1)-1 u(1) 1 cos(x), for any x ? ?-[-1,1]](http://img.homeworklib.com/questions/a93a8dc0-c0c8-11ea-a229-931d7671840b.png?x-oss-process=image/resize,w_560)
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Question 1) (3 Points) Consider the strong form below. Derive the weak form equation. The space...
solve problem #1 depending on the given information Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann b...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10,...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
We can expect the solution u(x,y) to be in the form X(x)Y(y). or I believe that...
We can expect the solution u(x,y) to be in the form
X(x)Y(y).
or
I believe that these are the correct forms of X(x) and Y(y).
2. Laplace's equation Consider Laplace's equation on the rectangle with 0 < x < L and 0 < < H: PDE BC BC BC u(x,0) 0, u(z, H) = g(z). (10) where a mixture of Dirichlet and Neumann boundary conditions is specified, and only one of the sides has a boundary condition that is nonhomogeneous...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variab...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in equation (1) derive the trigonometric form of Legendre equation for a function T (0) where 0 θ π: sin θ Then the general solution to (3) is T (0) y(cos θ) AP, (cos0) + BQ, (cos0).
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in...
Extra Credit Question (5 points): 1. Derive the Henderson-Hasselbach equation for weak acids. I have given...
Extra Credit Question (5 points): 1. Derive the Henderson-Hasselbach equation for weak acids. I have given you the equation below. ] pH = pk. - log [nonionized [ionized ] Start from here: K = [H*][A] [HA]
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the...
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t)...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
3. Consider the following game in normal form. Player 1 is the "row" player with strate-...
3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W Z X y a 3,3 2,1 0,2 2,1 b 1,1 1,2 1,0 1,4 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 с Find all the Nash equilibria in the game in pure strategies.
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y)...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
solve problem #1 depending on the given information
Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
We can expect the solution u(x,y) to be in the form
X(x)Y(y).
or
I believe that these are the correct forms of X(x) and Y(y).
2. Laplace's equation Consider Laplace's equation on the rectangle with 0 < x < L and 0 < < H: PDE BC BC BC u(x,0) 0, u(z, H) = g(z). (10) where a mixture of Dirichlet and Neumann boundary conditions is specified, and only one of the sides has a boundary condition that is nonhomogeneous...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in equation (1) derive the trigonometric form of Legendre equation for a function T (0) where 0 θ π: sin θ Then the general solution to (3) is T (0) y(cos θ) AP, (cos0) + BQ, (cos0).
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in...
Extra Credit Question (5 points): 1. Derive the Henderson-Hasselbach equation for weak acids. I have given you the equation below. ] pH = pk. - log [nonionized [ionized ] Start from here: K = [H*][A] [HA]
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W Z X y a 3,3 2,1 0,2 2,1 b 1,1 1,2 1,0 1,4 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 с Find all the Nash equilibria in the game in pure strategies.
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
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