Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: Partial survey, classification, new results, and generalizations

•Various exact decomposition methods are presented.•Decomposition methods of higher orders are described.•Equations of continuum mechanics and physics are considered.•More than fifteen coupled 3D equations are studied.•Considered methods permit one to construct exact solutions. The present paper provides a systematic treatment of various decomposition methods for linear (and some model nonlinear) systems of coupled three-dimensional partial differential equations of a fairly general form. Special cases of the systems considered are commonly used in applied mathematics, continuum mechanics, and physics. The methods in question are based on the decomposition (splitting) of a system of equations into a few simpler subsystems or independent equations. We show that in the absence of mass forces the solution of the system of four three-dimensional stationary and nonstationary equations considered can be expressed via solutions of three independent equations (two of which having a similar form) in a number of ways. The notion of decomposition order is introduced. Various decomposition methods of the first, second, and higher orders are described. To illustrate the capabilities of the methods, more than fifteen distinct systems of coupled 3D equations are discussed which describe viscoelastic incompressible fluids, compressible barotropic fluids, thermoelasticity, thermoviscoelasticity, electromagnetic fields, etc. The results obtained may be useful when constructing exact and numerical solutions of linear problems in continuum mechanics and physics as well as when testing numerical and approximate methods for linear and some nonlinear problems.