Generalized solutions of first order nonlinear Cauchy problems
Abstract
In this paper, we show how the Order Completion Method for systems of nonlinear partial differential equations may be applied to solve first order initial value problems. In particular, we construct generalized solutions of a large family of such initial value problems in two related spaces of generalized functions. The way in which the two mentioned solution concepts relate to each other is discussed, as well as the basic regularity properties of solutions.
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References
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[18] Sendov B., Hausdorff Approximations Kluwer Academic, Dordrecht, (1990).
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[23] Wyler Ein Komplettieringsfunktor für uniforme Limesräume, Mathematische Nacherichten 46 (1976), 1-12.
[2]Anguelov R., Markov S. and Sendov B., The set of Hausdorff continuous functions the largestlinear space of interval functions. Reliable Computing 12 (2006), 337-363.
[3]Anguelov R. and Rosinger E. E. Hausdorff continuous solution of nonlinear PDE through OCM. Quaestiones Mathematicae 28(3) (2005), 271-285.
[4] Arnold V. I. Lectures on PDEs, Springer Universitext, 2004.
[5] Baire R., Lecons sur les fonctions discontinues Collection Borel, Paris (1905).
[6] Barbu V. and Precupanu Th., Convexity and Optimization in Banach Spaces EdituraAcademiei, (1986).
[7] Beattie R. and Butzmann H.P., Convergence Struvtures and Application to Functional Analysis, Klummer Academic Publishers, (2002).
[8] Colombeau J. F. New generalized functions and multiplication of distributions, Noth Holland Mathematics Studies 84, 1984.
[9] Dilworth R. P., The normal completion of the lattice of continuous functions, AMS, Translatios, 68, (1950), 427-438.
[10] Gähler Grundstrukturen der Analysis II, Birkhäuser Verlag, Basel (1978).
[11] Kovalevskaia S. Zur Theorie der partiellen differentialgleichung, Journal für die reine und angewandte Mathematik 80 (1875) 1-32.
[12] Neuberger J. W. Sobolev Gradients and Differential; Equations Springer Lecture Notes in Mathematics, Springer Berlin, 1670, (1997).
[13] Neuberger J. W. Continuous Newton's method for polynimials Math. Intell. 21, (1999), 18 - 23.
[14] Neuberger J. W. A Near Minimal Hypothesis Nash-Moser Theorem, Int. J. Pure Appl. Math., 4, (2003), 269-280.
[15] Neuberger J. W. Prospects of a central theory of partial differential equations, Math. Intell., 27(3), (2005), 47-55.
[16] Oberguggenberger M. B., and Rosinger E. E. Solutions of continuous nonlinear PDEs through order completion North Holland, Amstardan (1994).
[17] Preuß G., Completion of semi-uniform convergence spaces, Appl. Categ. Structures, 8, (2000), 463-473.
[18] Sendov B., Hausdorff Approximations Kluwer Academic, Dordrecht, (1990).
[19] Van der walt, JH. The Order completion method for systems of nonlinear PDEs: Pseudotopological Perspectives. Acta Appl. Math. 103(2008), 1-17.
[20] Van der walt, JH. The uniform order convergence structure on ML(X) Quaest. Math. 31 (2008), 55-77.
[21] Van der walt, JH. The Order completion method for systems of nonlinear PDEs revisited. Acta Appl. Math. 106 (2009), 149-176.
[22] Van der walt, JH. The completion of Uniform Convergence Spaces and an Application to nonlinear PDEs Quaest. Math. 32 (2009), 371-395.
[23] Wyler Ein Komplettieringsfunktor für uniforme Limesräume, Mathematische Nacherichten 46 (1976), 1-12.
Published
2025-07-23
How to Cite
Agbebaku, D., Agbata, C., Ejikeme, C., & Okofu, M. (2025). Generalized solutions of first order nonlinear Cauchy problems. GPH-International Journal of Mathematics, 8(5), 32-54. https://doi.org/10.5281/zenodo.16355788
Section
Articles
Copyright (c) 2025 Dennis F. Agbebaku, Celestine B. Agbata, Chioma L. Ejikeme, Mary B. Okofu
The authors and co-authors warrant that the article is their original work, does not infringe any copyright, and has not been published elsewhere. By submitting the article to GPH - International Journal of Mathematics, the authors agree that the journal has the right to retract or remove the article in case of proven ethical misconduct.
