Advances in Higher-Order Linear Equations and Linear Polynomial Differential Operators

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 3291

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Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: mathematical analysis; applied mathematics; fractional calculus and its applications
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Special Issue Information

Dear Colleagues,

Linear equations and their nonlinear counterparts occur in many areas of mathematical, physical, biological, chemical, engineering, and statistical sciences. The theory and applications of the various tools and techniques of mathematical analysis, especially those that are based on linear polynomial differential operators, are being investigated widely and extensively in the current scientific literature. In this Special Issue, we welcome original as well as review-cum-expository research articles which focus on recent and new developments on the topics of higher-order linear and nonlinear equations and associated polynomial differential operators as well as their multidisciplinary applications.

We look forward to receiving and editorially processing your contributions to this Special Issue.

With kind regards and thank you in advance for your contributions.

Prof. Dr. Rekha Srivastava
Guest Editor

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Keywords

  • theory and applications of higher-order linear equations
  • theory and applications of systems of linear and nonlinear equations
  • applications based especially on linear polynomial differential operators
  • applications involving special functions and polynomials
  • applications based on other tools and techniques of mathematical analysis
  • theory and application of ordinary partial differential equations as well as their fractional-order relatives

Published Papers (4 papers)

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Research

16 pages, 11346 KiB  
Article
Laguerre-Type Bernoulli and Euler Numbers and Related Fractional Polynomials
by Paolo Emilio Ricci, Rekha Srivastava and Diego Caratelli
Mathematics 2024, 12(3), 381; https://doi.org/10.3390/math12030381 - 24 Jan 2024
Viewed by 683
Abstract
We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this [...] Read more.
We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0). Full article
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0 pages, 247 KiB  
Article
Compact Resolutions and Analyticity
by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López
Mathematics 2024, 12(2), 318; https://doi.org/10.3390/math12020318 - 18 Jan 2024
Viewed by 670
Abstract
We consider the large class G of locally convex spaces that includes, among others, the classes of (DF)-spaces and (LF)-spaces. For a space E in class G we have characterized that a subspace Y of [...] Read more.
We consider the large class G of locally convex spaces that includes, among others, the classes of (DF)-spaces and (LF)-spaces. For a space E in class G we have characterized that a subspace Y of (E,σ(E,E)), endowed with the induced topology, is analytic if and only if Y has a σ(E,E)-compact resolution and is contained in a σ(E,E)-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C(X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τp of C(X) is analytic iff (C(X),ξ) is separable and is covered by a compact resolution. Full article
21 pages, 2656 KiB  
Article
Insight into Functional Boiti–Leon–Mana–Pempinelli Equation and Error Control: Approximate Similarity Solutions
by Manal Alqhtani, Rekha Srivastava, Hamdy I. Abdel-Gawad, Jorge E. Macías-Díaz, Khaled M. Saad and Waleed M. Hamanah
Mathematics 2023, 11(22), 4569; https://doi.org/10.3390/math11224569 - 7 Nov 2023
Cited by 3 | Viewed by 715
Abstract
The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal [...] Read more.
The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal functions. In these cases, the F-BLMPE reduces to an explicit form in the dependent variable. In addition to this, it is worth deriving approximate similarity solutions of the F-BLMPE with constant coefficients using the extended unified method (EUM). In this method, nonlinear partial differential equation (NLPDE) solutions are expressed in polynomial and rational forms through an auxiliary function (AF) with adequate auxiliary equations. Exact solutions are estimated using formal solutions substituted into the NLPDEs, and the coefficients of the AF of all powers are set equal to zero. This approach is valid when the NLPDE is integrable. However, this technique is not valid for non-integrable equations, and only approximate solutions can be found. The maximum error can be controlled by an adequate choice of the parameters in the residue terms (RTs). Multiple similarity solutions are derived, and the ME is depicted in various examples within this work. The results found here confirm that the EUM is an efficient method for solving NLPDEs of the F-BLMPE type. Full article
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20 pages, 1155 KiB  
Article
Distributional Representation of a Special Fox–Wright Function with an Application
by Asifa Tassaddiq, Rekha Srivastava, Ruhaila Md Kasmani and Dalal Khalid Almutairi
Mathematics 2023, 11(15), 3372; https://doi.org/10.3390/math11153372 - 1 Aug 2023
Cited by 1 | Viewed by 784
Abstract
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta [...] Read more.
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution. Full article
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