Concluding Remarks
In concluding this book we give a brief con1mentary regarding the main
results forth in the foregoing. In order to make the book accessible not
only to specialists, but also to students and engineers, we give in Chapter 1
a complete account of definitions and notations and present a number of
relevant topics from other branches of mathen1atics. The detailed outline
of mathematical models leading to partial differential equations is available
in many textbooks and monographs on equations of mathematical physics.
In particular, we refer the reader to Courant and Hilbert (1953, 1962), Go-
dunov (1971), Morse and Feshbach (1953), Tikhonov and Samarskil (1963).
Current exposition follows the best legacies of the past: my first book with
1ny dear teacher - the late Academ. Tikhonov "Equations of Mathe1nat-
ical Physics", throughout which the reader can find thorough, advanced-
undergraduate to graduate level treatments of problems leading to partial
differential equations: hyperbolic, parabolic, elliptic equations; wave prop-
agations in space, heat conduction in space, special functions, etc. with
en1phasis on the rnathe1natical formulation of proble1ns, rigorous solutions,
physical interpretation of the results obtained.
The books by Gelfand ( 1967), Samarskil and Nikolaev (1989) cover in
full details the general theory of linear difference equations. Son1etimes the
elimination method available for solving various syste1ns of algebraic equa-
tions is referred to, in the foreign literature, as Thomas' algorithm and this
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