On representation of a solution to a modified Cauchy problem

Mamadaliev, N. K.

doi:10.1007/bf02674745

Cited by 14 publications

(4 citation statements)

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“…Indeed, even though many authors have addressed Cauchy problems for degenerate hyperbolic equations in two variables, most studies are restricted to equations where the 𝜕 2 𝜕𝑥 2 term vanishes at an initial line 𝑦 = 𝑦 0 (we refer to [15, §2.3], [149, Section 5.4] and references therein). Much less is known for hyperbolic equations whose 𝜕 2 𝜕𝑦 2 term vanishes at the same initial line: it is known that the Cauchy problem is, in general, not well-posed, and the relevance of determining conditions for its well-posedness has long been pointed out [15, §2.4], but as far as we are aware little progress has been made on this problem (for related work see [122]). The application of spectral techniques to hyperbolic Cauchy problems associated with Sturm-Liouville operators is by no means new, see e.g.…”

Section: The Associated Hyperbolic Cauchy Problemmentioning

confidence: 99%

Convolution-like Structures, Differential Operators and Diffusion Processes

Sousa¹,

Guerra²,

Yakubovich³

2022

Lecture Notes in Mathematics

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PhD Seminar; Maria José Moreira for her hospitality and all the help with the seminars; and my fellow students for their patience and friendship.There is much life beyond mathematics. To all my friends at the parish of São João da Madeira and at the movement JIM -Jovens em Missão: you are those who do not let me forget that life is more meaningful when our energy and skills are shared with the community. Thanks for the great moments we have shared during these years.A huge thanks goes to my parents Óscar and Cristina and to my uncle Augusto. The gratitude I feel for the way you support me and for everything you have teached me since the first day of my existence goes beyond anything that can be expressed in words.My PhD studies were made possible thanks to the financial support of the FCT -Fundação para a Ciência e a Tecnologia, I.P. through the PhD grants PD/BI/128072/2016 and PD/BD/135281/2017, and of CMUP under the project with reference UIDB/00144/2020, which is financed by FCT through national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020.

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Section: The Associated Hyperbolic Cauchy Problemmentioning

confidence: 99%

Convolution-like Structures, Differential Operators and Diffusion Processes

Sousa¹,

Guerra²,

Yakubovich³

2022

Lecture Notes in Mathematics

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“…Indeed, even though many authors have addressed Cauchy problems for degenerate hyperbolic equations in two variables, most studies are restricted to equations where the ∂ 2 ∂x 2 term vanishes at an initial line y = y 0 (we refer to [5, §2.3], [43,Section 5.4] and references therein). Much less is known for hyperbolic equations whose ∂ 2 ∂y 2 term vanishes at the same initial line: it is known that the Cauchy problem is, in general, not well-posed, and the relevance of determining sufficient conditions for its well-posedness has long been pointed out [5, §2.4], but as far as we are aware little progress has been made on this problem (for related work see [38]). The application of spectral techniques to hyperbolic Cauchy problems associated with Sturm-Liouville operators is by no means new, see e.g.…”

Section: Existence and Uniqueness Of Solutionmentioning

confidence: 99%

The hyperbolic maximum principle approach to the construction of generalized convolutions

Sousa¹,

Guerra²,

Yakubovich³

2019

Preprint

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We introduce a unified framework for the construction of convolutions and product formulas associated with a general class of regular and singular Sturm-Liouville boundary value problems. Our approach is based on the application of the Sturm-Liouville spectral theory to the study of the associated hyperbolic equation. As a by-product, an existence and uniqueness theorem for degenerate hyperbolic Cauchy problems with initial data at a parabolic line is established.The mapping properties of convolution operators generated by Sturm-Liouville operators are studied. Analogues of various notions and facts from probabilistic harmonic analysis are developed on the convolution measure algebra. Various examples are presented which show that many known convolution-type operators -including those associated with the Hankel, Jacobi and index Whittaker integral transforms -can be constructed using this general approach.

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“…The Cauchy problem for hyperbolic partial differential equations was considered by other methods as well, mainly, by methods of functional analysis. But they do not permit one to construct solutions in analytic form, and only the well-posed statement of the problem can be proved [6][7][8][9][10][11][12][13][14][15]. Note also the paper [16] in which analytic forms of the general solution for partial differential equations and solutions of the Cauchy problem were constructed with the use of computer algebra systems.…”

Section: Introductionmentioning

confidence: 99%