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1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t): PEx) at (a) (10 ...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0,...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determi...
#2 ONLY PLEASE
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx +...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
We can apply the Method of Separation of Variables to obtain a representation for the solution u ...
Please show all work and provide and an original solution.
We can apply the Method of Separation of Variables to obtain a representation for the solution u u(, t) for the following partial differential equation (PDE) on a bounded domain with homogeneous boundary conditions. The PDE model is given by: u(r, 0) 0, (2,0) = 4. u(0,t)0, t 0 t 0 (a) (20 points) Assume that the solution to this PDE model has the form u(x,t) -X (r) T(t). State...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and...
3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and σ(z) are real and continous on la, b , and p(2) and σ(z) are strictly positive for all a,b (a) Derive the Rayleigh quotient λ from (2). b) What does this quotient describe? Give two examples of applications for this formula. (c) what are the neces,ary conditions for λ > 0 to be satisfied? (d) Recall that the minimum value of the Rayleigh quotient...
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-o...
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-order differential equations of the type is called generalized Riccati equations. It is another nonlinear ordinary differential equation (a) Suppose, P2 differentiable and P2メ0 on I. By using the Ansatz u(z) :-y(r) P2(x) T, for every z where y is a solution of (2), develope a method to solve the equation (2). Describe in brief steps your method. Hint: The...
Consider the linear second-order PDE for u = u(x, y), 2uxx – 3uxy – 2uyy =...
Consider the linear second-order PDE for u = u(x, y), 2uxx – 3uxy – 2uyy = (2x + y)2. (i) Determine the type (elliptic, hyperbolic, or parabolic) of (*). (ii) Introduce new independent variables s, t via x = 8 + 2t, y = -2s+t, and let w = w(s, t) be the function u in these new variables, i.e., let w(s, t) = u(s + 2t, -2s +t). Utilizing the chain rule, Ug = W58x + witz, Uy =...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt + Kurz = 0 for 0 < x <L (30) where K > 0 is a constant. Suppose the boundary conditions are given by (31) u(0, t) = uz(0,t) = 0 Uwx (L, t) = Uzzz(L, t) = 0 (32) and the initial conditions are (33) u(x,0) = (x) u1(x,0) = V(x) (34) Use separation of variables to find the general solution to the...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
#2 ONLY PLEASE
1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
Please show all work and provide and an original solution.
We can apply the Method of Separation of Variables to obtain a representation for the solution u u(, t) for the following partial differential equation (PDE) on a bounded domain with homogeneous boundary conditions. The PDE model is given by: u(r, 0) 0, (2,0) = 4. u(0,t)0, t 0 t 0 (a) (20 points) Assume that the solution to this PDE model has the form u(x,t) -X (r) T(t). State...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and σ(z) are real and continous on la, b , and p(2) and σ(z) are strictly positive for all a,b (a) Derive the Rayleigh quotient λ from (2). b) What does this quotient describe? Give two examples of applications for this formula. (c) what are the neces,ary conditions for λ > 0 to be satisfied? (d) Recall that the minimum value of the Rayleigh quotient...
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-order differential equations of the type is called generalized Riccati equations. It is another nonlinear ordinary differential equation (a) Suppose, P2 differentiable and P2メ0 on I. By using the Ansatz u(z) :-y(r) P2(x) T, for every z where y is a solution of (2), develope a method to solve the equation (2). Describe in brief steps your method. Hint: The...
Consider the linear second-order PDE for u = u(x, y), 2uxx – 3uxy – 2uyy = (2x + y)2. (i) Determine the type (elliptic, hyperbolic, or parabolic) of (*). (ii) Introduce new independent variables s, t via x = 8 + 2t, y = -2s+t, and let w = w(s, t) be the function u in these new variables, i.e., let w(s, t) = u(s + 2t, -2s +t). Utilizing the chain rule, Ug = W58x + witz, Uy =...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt + Kurz = 0 for 0 < x <L (30) where K > 0 is a constant. Suppose the boundary conditions are given by (31) u(0, t) = uz(0,t) = 0 Uwx (L, t) = Uzzz(L, t) = 0 (32) and the initial conditions are (33) u(x,0) = (x) u1(x,0) = V(x) (34) Use separation of variables to find the general solution to the...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
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