Fundamental Solutions of Linear Differential Equations in the Sense of Mnemofunctions
Tóm tắt
In this paper we discuss the definition and construction of the fundamental solution of a linear differential equation with constant coefficients in the algebra of mnemofunctions. This algebra contains the tempered distributions. We also present some examples including a physical illustration of our results for the equation Δu = δ.
Tài liệu tham khảo
A. B. Antonevich and Ya V. Radyno, On a general method of construction of generalized functions, Dokl. Acad. Nauk USSR 318, 267-270 (1991).
N. N. Bogolubov and D. V. Shirkov, Introduction to Quantum Field Theory, Nauka, Moscow, 1976.
J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland, Amsterdam, 1985.
Yu V. Egorov, On the theory of generalized functions, Usp. Mat. Nauk 45, 3-40 (1990).
L. Ehrenpries, Solutions of some problems of division, Amer. J. Math. 76, 883-903 (1954).
L. Hörmander, On the division of distributions by polynomials, Ark. Mat. 6, 555-568 (1958).
A. N. Krylov, Differential Equations of Mathematical Physics, Academy of Sciences USSR, Leningrad, 1933.
B. Malgrange, Equations aux dériverées partielles à coefficients constants, 1. Solution élémentair, C. R. Acad. Sci. (Pris) 237, 1620-1622 (1953).
Ya V. Radyno and Fu Than Ngo, The differential equations in the algebra of the new generalized functions, Dokl. Acad. N. Belarusi 37, 5-9 (1993).
Ya V. Radyno, Fu Than Ngo, and Sabra Ramadan, Fourier transformation in the algebra of new generalized functions, Dokl. Ross. Acad. Nauk 327, 20-24 (1992).
L. Schwartz, Theorie des Distributions, Vols. 1 and 2, Paris, 1950–1951.
L. Schwartz, Sur l'impossibilitie de la multiplication des distributions, C. R. Acad. Sci. (Paris) 239, 847-848 (1954).
F. Trèves, Solution élémentaire d'équations aux dériveés partielles dépend d'un paramètre, C. R. Acad. Sci. (Paris) 242, 1250-1252 (1956).