A temperature problem for a square: An exact solution

We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.

An inhomogeneous problem for an elastic half-strip: An exact solution

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Existence of Solutions

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The discussion centers on the solution w = v + u, where v solves the homogeneous Cauchy problem with v(x, 0) = f(x) and u solves the inhomogeneous problem with u(x, 0) = 0. The chapter first provides definitions for the Wright–Fisher–Hölder spaces on a general compact manifold with corners before explaining the steps involved in the existence proof. It then verifies the induction hypothesis and treats the k = 0 case. It also shows how to perform the doubling construction for P and considers the existence of the resolvent operator and a contraction semi-group. Finally, it discusses the problem of higher regularity.

Analytical Solution of Unsteady State Conduction Problem with Heat Source

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.

Basis-set methods for the Dirac equation

The pathologies associated with finite basis-set approximations to the Dirac Hamiltonian HDirac are avoided by applying the variational principle to the bounded operator 1 / (H Dirac – W) where W is a real number that is not in the spectrum of HDirac. Methods of calculating upper and lower bounds to eigenvalues, and bounds to the wave-function error as measured by the L2 norm, are described. Convergence is proven. The rate of convergence is analyzed. Boundary conditions are discussed. Benchmark energies and expectation values for the Yukawa potential, and for the Coulomb plus Yukawa potential, are tabulated. The convergence behavior of the energy-weighted dipole sum rules, which have traditionally been used to assess the quality of basis sets, and the convergence behavior of the solutions to the inhomogeneous problem, are analyzed analytically and explored numerically. It is shown that a basis set that exhibits rapid convergence when used to evaluate energy-weighted dipole sum rules can nevertheless exhibit slow convergence when used to solve the inhomogeneous problem and calculate a polarizability. A numerically stable method for constructing projection operators, and projections of the Hamiltonian, onto positive and negative energy states is given. PACS Nos.: 31.15Pf, 31.30Jv, 31.15-p