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Basis functions and weak ODE solution

proving Orthonormal basisSolving differential equation by weak formulation and minimizing a functionalWeak topologies and weak convergence - Looking for feedbacksSimple coupled ODEOperator Convergence in Hilbert spaceEigenvalue of every eigenvector is an eigenvalue of element of o.n eigenvector basisBasis of $H_0^1$ of solutions to the time-independent Schrödinger equationFinding Orthonormal Basis from Orthogonal BasisWhich is the partial solution of the ode 4th orderGeneral solutions to differential equations and loss of information about eigenvalues

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$begingroup$

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = lambda u$. Given basis functions, call them $phi_i$, I use a variational procedure and the Ritz method to approximate $lambda$ via the associated weak formulation
$$langle L(phi_i),phi_jrangle = lambda langle phi_i,phi_jrangle.$$

As you can see, this expression is now a matrix equation, solutions to which are straightforward. For my particular problem, the basis functions are $$phi_j = cosleft( frac{pi j}{2}(x+1) right) coshleft( frac{pi j}{2}(y+h) right).$$

However, this solution, when inputted into the weak formulation equation, does not output correct eigenvalues. However, $phi_j$ can be split into even and odd components:
$$ phi_j^o = sin left( pi(j-1/2)x right)coshleft( pi(p-1/2)(y+h) right)\
phi_j^e = cos left( pi j x right)coshleft( pi j(y+h) right)
$$

Now to obtain eigenvalues I solve two separate equations, one for even eigenvalues and one for odd:
$$langle L(phi_i^e),phi_j^erangle = lambda langle phi_i^e,phi_j^erangle\
langle L(phi_i^o),phi_j^orangle = lambda langle phi_i^o,phi_j^orangle.$$

This latter approach gives correct solutions: why? Any insight or direction is greatly appreciated.

functional-analysis ordinary-differential-equations change-of-basis

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asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

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0

$begingroup$

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = lambda u$. Given basis functions, call them $phi_i$, I use a variational procedure and the Ritz method to approximate $lambda$ via the associated weak formulation
$$langle L(phi_i),phi_jrangle = lambda langle phi_i,phi_jrangle.$$

As you can see, this expression is now a matrix equation, solutions to which are straightforward. For my particular problem, the basis functions are $$phi_j = cosleft( frac{pi j}{2}(x+1) right) coshleft( frac{pi j}{2}(y+h) right).$$

However, this solution, when inputted into the weak formulation equation, does not output correct eigenvalues. However, $phi_j$ can be split into even and odd components:
$$ phi_j^o = sin left( pi(j-1/2)x right)coshleft( pi(p-1/2)(y+h) right)\
phi_j^e = cos left( pi j x right)coshleft( pi j(y+h) right)
$$

Now to obtain eigenvalues I solve two separate equations, one for even eigenvalues and one for odd:
$$langle L(phi_i^e),phi_j^erangle = lambda langle phi_i^e,phi_j^erangle\
langle L(phi_i^o),phi_j^orangle = lambda langle phi_i^o,phi_j^orangle.$$

This latter approach gives correct solutions: why? Any insight or direction is greatly appreciated.

functional-analysis ordinary-differential-equations change-of-basis

share|cite|improve this question

asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = lambda u$. Given basis functions, call them $phi_i$, I use a variational procedure and the Ritz method to approximate $lambda$ via the associated weak formulation
$$langle L(phi_i),phi_jrangle = lambda langle phi_i,phi_jrangle.$$

As you can see, this expression is now a matrix equation, solutions to which are straightforward. For my particular problem, the basis functions are $$phi_j = cosleft( frac{pi j}{2}(x+1) right) coshleft( frac{pi j}{2}(y+h) right).$$

However, this solution, when inputted into the weak formulation equation, does not output correct eigenvalues. However, $phi_j$ can be split into even and odd components:
$$ phi_j^o = sin left( pi(j-1/2)x right)coshleft( pi(p-1/2)(y+h) right)\
phi_j^e = cos left( pi j x right)coshleft( pi j(y+h) right)
$$

Now to obtain eigenvalues I solve two separate equations, one for even eigenvalues and one for odd:
$$langle L(phi_i^e),phi_j^erangle = lambda langle phi_i^e,phi_j^erangle\
langle L(phi_i^o),phi_j^orangle = lambda langle phi_i^o,phi_j^orangle.$$

This latter approach gives correct solutions: why? Any insight or direction is greatly appreciated.

functional-analysis ordinary-differential-equations change-of-basis

share|cite|improve this question

asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

Josh McCraneyJosh McCraney

11

New contributor

Josh McCraney is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

Josh McCraneyJosh McCraney

11