zbMATH — the first resource for mathematics

Division problem and partial differential equations with constant coefficients in Colombeau’s space of new generalized functions. (English) Zbl 0868.35017
Summary: We study the problem of division in Colombeau’s space of new generalized functions in the associated sense: more precisely we solve the equation of the form \(F\cdot G\approx_pH\) with adequate assumptions on \(F\). By using the Fourier transformation we construct, by simple methods, the solution in the \(p\)-associated sense of a partial differential equation with constant coefficients.

35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
26C10 Real polynomials: location of zeros
46F05 Topological linear spaces of test functions, distributions and ultradistributions
Full Text: DOI EuDML
[1] Biagioni, H. A.: A Nonlinear Theory of Generalized Functions. Lect. Notes Math. 1421. Berlin-Heidelberg-New York: Springer. 1990. · Zbl 0694.46032
[2] Colombeau, J. F.: Elementary Introduction to New Generalized Functions. Amsterdam: North-Holland. 1985. · Zbl 0584.46024
[3] Colombeau, J. F.: Multiplication of distributions. Bull. Amer. Math. Soc.23, 251–268 (1990). · Zbl 0731.46023
[4] Egorov, J. V.: Theory of generalized functions. Uspehi Math. Nauk45, 1–40 (1990). · Zbl 0754.46034
[5] Hörmander, L.: On the division of distributions by polynomials. Arkiv Mat.3, 555–568 (1958). · Zbl 0131.11903
[6] Lojasiewicz, S.: Sur le problem de division. Stud. Math.18, 87–136 (1959).
[7] Nedeljkov, M., Pilipović, S.: Convolution in Colombeau’s spaces of generalized functions. I and II. Publ. Inst. Math. (N. S.)52, 95–105; 105–113 (1992). · Zbl 0790.46025
[8] Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. New York: Longman. 1992. · Zbl 0818.46036
[9] Pilipović, S.: Colombeau’s Generalized Functions and Pseudo-differential operators. Lect. Notes Math. Sci. 4. Univ. of Tokio. 1994.
[10] Rosinger, E. E.: Generalized Solutions to Nonlinear Partial Differential Equations. Amsterdam: North-Holland. 1987. · Zbl 0635.46033
[11] Scarpalézos, D.: Colombeau’s generalized functions, topological structures, microlocal properties. A simplified point of view. Preprint. · Zbl 1011.46042
[12] Van der Warden, L.: Algebra Vol. I, II. Berlin-Heidelberg-New York: Springer. 1971, 1967
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.