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A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical biology, and the study of porous media.

We present a population of 56 linear, two-dimensional elliptic partial differential equations (PDEs) suitable for evaluating numerical methods and software. Forty two of the PDEs are parameterized which allows much larger populations to be made; 189 specific cases are presented here along with solutions (some are only approximate). Many of the PDEs are artificially created so as to exhibit various mathematical behaviors of interest; the others are taken from real world problems in various ways. The population has been structured by introducing measures of complexity of the operator, boundary conditions, solution, and problem. The PDEs are first presented in mathematical terms along with contour plots of the 189 specific solutions. Machine readable descriptions are given in Part 2, MRC Technical Summary Report no. 2079; many of the PDEs involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions.

This report contains the machine readable form of the 189 partial differential equations presented in MRC Technical Summary Report No. 2078 'A POPULATION OF LINEAR, SECOND ORDER, ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS ON RECTANGLES - PART 1.' These are two large files: EQNFIL contains 189 entries which are either complete statements of a PDE in the ELLPACK language or a reference to an entry in MACFIL with parameter values given. MACFIL contains the 42 parameterized PDE's in a form suitable for automatically substituting in parameter values. A few other small files are given to facilitate using this problem population within the ELLPACK system. (Author)