Differential operator, self-adjoint

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

Clearly, stability is an intrinsic property of schemes regardless of approximations and interrelations between the resulting schemes and relevant differential equations. Because of this, any stability condition should be imposed as the relationship between the operators A and B. More specifically, let a family of schemes specified by the restrictions on the operators A and B be given A = A > 0 or Ay, v) = y, Av) and Ay, y) > 0 for any y, v H, where (, ) is an inner product in H, B > 0 and B B B is non-self-adjoint). The problem statement consists of extracting from that family a set of schemes that are stable with respect to the initial data, having the form... 

The domain of the Schrodinger operator on the graph is the L2 space of differentiable functions which are continuous at the vertices. The operator is constructed in the following way. On the bonds, it is identified as the one dimensional Laplacian — It is supplemented by boundary conditions on the vertices which ensure that the resulting operator is self adjoint. We shall consider in this paper the Neumann boundary conditions ... 

The establishment of difference equations is based on the discretisation of the self-adjoint differential operator... 

The application of the Sturm-Liouville integral transform using the general linear differential operator (11.45) has now been demonstrated. One of the important new components of this analysis is the self-adjoint property defined in Eq. 11.50. The linear differential operator is then called a self-adjoint dijferential operator. 

Before we apply the Sturm-Liouville integral transform to practical problems, we should inspect the self-adjoint property more carefully. Even when the linear differential operator (Eq. 11.45) possesses self-adjointness, the self-adjoint property is not complete since it actually depends on the type of boundary conditions applied. The homogeneous boundary condition operators, defined in Eq. 11.46, are fairly general and they lead naturally to the self-adjoint property. This self-adjoint property is only correct when the boundary conditions are unmixed as defined in Eq. 11.46, that is, conditions at one end do not involve the conditions at the other end. If the boundary conditions are mixed, then the self-adjoint property may not be applicable. 

The next important property we need to prove is the self-adjoint property for the differential operator defined in Eq. 11.188. Let u and v be elements of the function space, i.e., they must satisfy the boundary values... 

The RHS is the definition of . Thus, we have proved the self-adjoint property for the differential operator L. 

Finite models. Because of their mathematical simplicity, theoretical models of the types described above are those most frequently analyzed in detail in the reactor literature. However, the self-adjoint nature of the differential operators postulated is not typical of real reactor problems neither is the symmetry of the associated integral kernels. This is because, physically, neutron life-histories are not reversible in heterogeneous reactors, even statistically the optical reciprocity theorem [4, p. 82] is not vaUd when slowing down is considered. 

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