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The Darboux problem and the definition of the Riemann–Hadamard function are presented for the Bianchi equation of the fourth order—a linear fourth-order equation with four independent variables that has a dominant derivative not containing multiple differentiation with respect to any of the independent variables. Sufficient conditions on the coefficients of this equation under which its Riemann–Hadamard function allows explicit construction in terms of hypergeometric functions are obtained.

For a linear inhomogeneous system of first-order partial differential equations with three independent variables, we prove the existence and uniqueness of a solution of the Darboux problem, which is constructed in terms of the Riemann–Hadamard matrix defined in the paper.

We prove the existence and uniqueness of a solution of the Darboux problem for the fourth-order Bianchi equation. The Riemann–Hadamard function is determined for the Darboux problem. The solution of the problem is constructed in terms of this function.

Laplace invariants are constructed for a fourth-order equation that is a generalization of the Hallaire equation. The determining equations are written in terms of these invariants.

For a system of three first-order partial differential equations with three independent variables, we obtain sufficient conditions for one component of the solution to satisfy a third-order Bianchi equation. We also obtain conditions for the solvability of this system by quadratures.

We construct Laplace invariants for an equation with a dominating fourth partial derivative. The determining equations are written out in terms of the Laplace invariants. We single out classes of equations admitting four-dimensional Lie algebras.

On the basis of defining equations written out in terms of Laplace invariants, we single out some classes of fourth-order Bianchi equations similar to the well-known classes of hyperbolic equations with two independent variables admitting Lie algebras of maximum dimension.

For an equation with four independent variables, we suggest conditions on the coefficients under which the Riemann function is a solution of a system of two integral equations. The result is used to construct the Riemann function in closed form.

We construct the Laplace invariants of the Bianchi equation with four independent variables and obtain determining equations expressed in terms of these invariants.

in the Euclidean space R, where α = (α1, α2, α3, α4) = (2, 1, 1, 1), the multi-indices β have four components, and we write β < α if β is obtained from α by diminishing at least one component. This equation belongs to the class of equations of the form (D +M)u = f (x1, . . . , xn) , D ≡ D, α = (m1,m2, . . . ,mn) , (2) where M is a linear differential operator with variable coefficients that contains only derivatives obtained from D by omitting at least one differentiation. Formulas for the solution of the Cauchy problem were obtained in [1, 2] in terms of the Riemann function in the threeand four-dimensional Euclidean spaces for equations of the form (1) with α = (1, 1, . . . , 1). In [3], a similar result was obtained for equations of the form (2) in R with α = (2, 1) and in R with α = (2, 1, 1). Consider Eq. (1) with coefficients aβ ∈ C and right-hand side f ∈ C. The class C, β = (β1, β2, β3, β4), contains functions such that all of their derivatives D, γ = (γ1, γ2, γ3, γ4), γi = 0, . . . , βi, i = 1, 2, 3, 4, exist and are continuous. The Riemann function R (x1, x2, x3, x4; ξ1, ξ2, ξ3, ξ4) for Eq. (1) is defined as a solution of the integral equation

Original Russian Text Copyright c © 2001 by Mironov. SHORT COMMUNICATIONS The Construction of the Riemann Function for a Fourth-Order Equation A. N. Mironov Kazan State University, Kazan, Russia Received July 24, 2001 The Riemann function v (x, y, z, t, x0, y0, z0, t0) of the equation L(u) = uxyzt + auxyz + buxyt + cuxzt + duyzt + euxy + fuxz + guxt + huyz + kuyt + suzt +mux + nuy + puz + qut + ru = 0 (1) was defined in [1] as a solution of the integral equation v(x, y, z, t) − t ∫ t0 (av)(x, y, z, δ)dδ − z ∫ z0 (bv)(x, y, γ, t)dγ − y ∫ y0 (cv)(x, β, z, t)dβ − x ∫ x0 (dv)(α, y, z, t)dα + z ∫ z0 t ∫ t0 (ev)(x, y, γ, δ)dδ dγ + y ∫ y0 t ∫ t0 (fv)(x, β, z, δ)dδ dβ + y ∫ y0 z ∫ z0 (gv)(x, β, γ, t)dγ dβ + x ∫ x0 t ∫ t0 (hv)(α, y, z, δ)dδ dα+ x ∫ x0 z ∫ z0 (kv)(α, y, γ, t)dγ dα + x ∫ x0 y ∫ y0 (sv)(α, β, z, t)dβ dα− y ∫ y0 z ∫ z0 t ∫ t0 (mv)(x, β, γ, δ)dδ dγ dβ − x ∫ x0 z ∫ z0 t ∫ t0 (nv)(α, y, γ, δ)dδ dγ dα− x ∫ x0 y ∫ y0 t ∫ t0 (pv)(α, β, z, δ)dδ dβ dα − x ∫ x0 y ∫ y0 z ∫ z0 (qv)(α, β, γ, t)dγ dβ dα+ x ∫ x0 y ∫ y0 z ∫ z0 t ∫ t0 (rv)(α, β, γ, δ)dδ dγ dβ dα = 1. (2)