Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation

The paper is devoted to the construction and study of the additive averaged semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. The studied equation is based on the well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

Hertz, Heinrich Rudolf (1857–94)

Heinrich Hertz demonstrated the existence of radio waves in research between 1887 and 1888, opening the way for Marconi to develop long-distance radio communication. Hertz’s results confirmed Maxwell’s electromagnetic theory, and sealed the fate of action-at-a-distance in physics. His theoretical analysis included the famous dictum: ‘Maxwell’s theory is Maxwell’s system of equations’. Hertz also developed a new formulation of Newtonian mechanics using the concepts of mass, length and time, but not force. He presented mechanics as the axiomatic consequence of a single fundamental law: ‘every free system persists in its state of rest or of uniform motion in a straightest path’. Hertz’s ideas influenced later philosophers of science but were most important as a source for Wittgenstein’s Tractatus Logico-Philosophicus, which influenced logical positivism. Hertz’s proposal to eliminate the concept of force in physics was an important contribution to the twentieth-century ideal of a philosophical method that does not solve, but rather dissolves, philosophical problems.