Items related to Integral Operators in the Theory of Linear Partial...
Synopsis
I. Differential equations in two variables with entire coefficients.- § 1. A representation of solutions of partial differential equations.- § 2. The integral operator of the first kind.- § 3. Further representations of integral operators.- § 4. A representation of the operator of the first kind in terms of integrals.- § 5. Properties of the integral operator of the first kind.- § 6. Some further properties of the integral operator of the first kind.- § 7. The differential equation ?2V + F(r2) V = 0.- § 8. Integral operators of exponential type.- § 9. The differential equation ?2? + N (x) ? = 0.- § 10. Differential equations of higher order.- II. Harmonic functions in three variables.- § 1. Preliminaries.- § 2. Characteristic space ?3.- § 3. Harmonic functions with rational B3-associates.- § 4. Period functions.- § 5. Relations between coefficients of a series development of a harmonic function and its singularities.- § 6. Another type of integral representations of harmonic functions.- § 7. The behavior in the large of functions of the class S (E, ?0, ?1) with a rational associate f (?).- III. Differential equations in three variables.- § 1. An integral operator generating solutions of the equation ?3? + A (r2) X · ? ? + C (r2) ? = 0.- § 2. A series expansion for solutions of the equation ?3? + A (r2) X · ? ? + C (r2) ? = 0.- § 3. An integral operator generating solutions of the equation ?3? + F (y, z) ? = 0.- § 4. A second integral operator generating solutions of the equation ?3? + F (y, z) ? = 0.- § 5. An integral operator generating solutions of the equation ?x + ?yy + ?zz + F (y, z) ? = 0.- § 6. An integral operator generating solutions of the equation g???????+h????+k? = 0.- IV. Systems of differential equations.- § 1. Harmonic vectors of three variables. Preliminaries.- § 2. Harmonic vectors in the large and their representation as integrals.- § 3. Integrals of harmonic vectors.- § 4. Relations between integrals of algebraic harmonic vectors in three variables and integrals of algebraic functions of a complex variable.- § 5. Generalization of the residue theorems to the case of the equation ?3? + F (r2) ? = 0.- § 6. An operator generating solutions of a system of partial differential equations.- V. Equations of mixed type and elliptic equations with singular and non-analytic.- § 1. Introduction. The simplified case of equations of mixed type.- § 2. A generalization of the representation (1.12) of solutions of the equation (1.6).- § 3. The operator (1.11b) in the general case.- § 4. Generating functions analogous to solutions of the hypergeometric equation.- § 5. On the solution of the initial value problem in the large.- § 6. Generalized Cauchy-Riemann equations.- § 7. The differential equation ?2? + N(x)? = 0 with a new type of singularity of N.- § 8. An integral operator for equations with non-analytic coefficients.
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