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Difference Equations, Special Functions and Orthogonal Polynomials

Special functions and orthogonal polynomials in particular have been around for centuries. In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has been extended to difference equations, partial differential equations and non-linear differential equations. 

This special issue will reflect the interplay of difference/differential equations, special functions and polynomials. 

Guest Editors: Serkan Araci, H. M. Srivastava, Praveen Agarwal, and Juan Luis García Guirao

  1. We present unified versions of Minkowski-type fractional integral inequalities with the help of fractional integral operator based on a unified Mittag-Leffler function. These inequalities provide new as well a...

    Authors: Shuang-Shuang Zhou, Ghulam Farid and Ayyaz Ahmad
    Citation: Advances in Continuous and Discrete Models 2022 2022:9
  2. In this paper, the mathematical models for flow and heat-transfer analysis of a non-Newtonian fluid with axisymmetric channels and porous walls are analyzed. The governing equations of the problem are derived ...

    Authors: Naveed Ahmad Khan, Muhammad Sulaiman, Poom Kumam and Fawaz Khaled Alarfaj
    Citation: Advances in Continuous and Discrete Models 2022 2022:7
  3. In this paper, we introduce the concept of lacunary statistical boundedness of Δ-measurable real-valued functions on an arbitrary time scale. We also give the relations between statistical boundedness and lacu...

    Authors: Bayram Sözbir and Selma Altundağ
    Citation: Advances in Difference Equations 2021 2021:513
  4. The present paper introduces a new modification of Gamma operators that protects polynomials in the sense of the Bohman–Korovkin theorem. In order to examine their approximation behaviours, the approximation p...

    Authors: Reyhan Özçelik, Emrah Evren Kara, Fuat Usta and Khursheed J. Ansari
    Citation: Advances in Difference Equations 2021 2021:508
  5. In this paper, we construct a method with eight steps that belongs to the family of Obrechkoff methods. Due to the explicit nature of the new method, not only does it not require another method as predictor, b...

    Authors: Ali Shokri, Higinio Ramos, Mohammad Mehdizadeh Khalsaraei, Fikret A. Aliev and Martin Bohner
    Citation: Advances in Difference Equations 2021 2021:506
  6. In this paper, we define framed slant helices and give a necessary and sufficient condition for them in three-dimensional Euclidean space. Then, we introduce the spherical images of a framed curve. Also, we ex...

    Authors: Osman Zeki Okuyucu and Mevlüt Canbirdi
    Citation: Advances in Difference Equations 2021 2021:504
  7. The impact of Newtonian heating on a time-dependent fractional magnetohydrodynamic (MHD) Maxwell fluid over an unbounded upright plate is investigated. The equations for heat, mass and momentum are established...

    Authors: Nazish Iftikhar, Fatima Javed, Muhammad Bilal Riaz, Muhammad Abbas, Abdullah M. Alsharif and Y. S. Hamed
    Citation: Advances in Difference Equations 2021 2021:501
  8. In this paper a new approach is taken to find the exact solutions for generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate and is analyzed for ramped wall tem...

    Authors: Muhammad Bilal Riaz, Jan Awrejcewicz, Aziz Ur Rehman and Muhammad Abbas
    Citation: Advances in Difference Equations 2021 2021:500
  9. Two new iterative methods for the simultaneous determination of all multiple as well as distinct roots of nonlinear polynomial equation are established, using two suitable corrections to achieve a very high co...

    Authors: Mudassir Shams, Naila Rafiq, Nasreen Kausar, Praveen Agarwal, Choonkil Park and Shaher Momani
    Citation: Advances in Difference Equations 2021 2021:495
  10. The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fra...

    Authors: Tariq A. Aljaaidi, Deepak B. Pachpatte, Thabet Abdeljawad, Mohammed S. Abdo, Mohammed A. Almalahi and Saleh S. Redhwan
    Citation: Advances in Difference Equations 2021 2021:493
  11. Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, t...

    Authors: R. Sadat, Praveen Agarwal, R. Saleh and Mohamed R. Ali
    Citation: Advances in Difference Equations 2021 2021:486
  12. In this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations...

    Authors: Mudassir Shams, Naila Rafiq, Nasreen Kausar, Praveen Agarwal, Choonkil Park and Nazir Ahmad Mir
    Citation: Advances in Difference Equations 2021 2021:480
  13. In this paper, we study the uniqueness and existence of the solution of a non-autonomous and nonsingular delay difference equation using the well-known principle of contraction from fixed point theory. Further...

    Authors: Gul Rahmat, Atta Ullah, Aziz Ur Rahman, Muhammad Sarwar, Thabet Abdeljawad and Aiman Mukheimer
    Citation: Advances in Difference Equations 2021 2021:474
  14. In this paper, we establish certain new subclasses of meromorphic harmonic functions using the principles of q-derivative operator. We obtain new criteria of sense preserving and univalency. We also address other...

    Authors: Neelam Khan, H. M. Srivastava, Ayesha Rafiq, Muhammad Arif and Sama Arjika
    Citation: Advances in Difference Equations 2021 2021:471
  15. A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous...

    Authors: Mudassir Shams, Naila Rafiq, Nasreen Kausar, Praveen Agarwal, Choonkil Park and Nazir Ahmad Mir
    Citation: Advances in Difference Equations 2021 2021:465
  16. The central purpose of this effort is to investigate analytic and geometric properties of a class of normalized analytic functions in the open unit disk involving Bernoulli’s formula. As a consequence, some so...

    Authors: Rabha W. Ibrahim, Ibtisam Aldawish and Dumitru Baleanu
    Citation: Advances in Difference Equations 2021 2021:463
  17. The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. A...

    Authors: Tahereh Eftekhari, Jalil Rashidinia and Khosrow Maleknejad
    Citation: Advances in Difference Equations 2021 2021:461
  18. Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, ...

    Authors: Sidra Maqsood, Muhammad Abbas, Kenjiro T. Miura, Abdul Majeed, Gang Hu and Tahir Nazir
    Citation: Advances in Difference Equations 2021 2021:459
  19. In this paper, for the first time the inverse problem of reconstructing the time-dependent potential (TDP) and displacement distribution in the hyperbolic problem with periodic boundary conditions (BCs) and no...

    Authors: M. J. Huntul, Muhammad Abbas and Dumitru Baleanu
    Citation: Advances in Difference Equations 2021 2021:452
  20. In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and general source terms. According to some suitable hypothesis, we study the blow-up of solutions. This is the ...

    Authors: Salah Boulaaras, Abdelbaki Choucha, Praveen Agarwal, Mohamed Abdalla and Sahar Ahmed Idris
    Citation: Advances in Difference Equations 2021 2021:446
  21. This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various t...

    Authors: Shrideh Al-Omari, Dayalal Suthar and Serkan Araci
    Citation: Advances in Difference Equations 2021 2021:441
  22. In the present investigation, with the help of certain higher-order q-derivatives, some new subclasses of multivalent q-starlike functions which are associated with the Janowski functions are defined. Then, certa...

    Authors: Bilal Khan, Zhi-Guo Liu, H. M. Srivastava, Serkan Araci, Nazar Khan and Qazi Zahoor Ahmad
    Citation: Advances in Difference Equations 2021 2021:440
  23. The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomial...

    Authors: Taekyun Kim, Dmitry V. Dolgy, Dae San Kim, Hye Kyung Kim and Seong Ho Park
    Citation: Advances in Difference Equations 2021 2021:421
  24. A number of mathematical methods have been developed to determine the complex rheological behavior of fluid’s models. Such mathematical models are investigated using statistical, empirical, analytical, and ite...

    Authors: Muhammad Bilal Riaz, Kashif Ali Abro, Khadijah M. Abualnaja, Ali Akgül, Aziz Ur Rehman, Muhammad Abbas and Y. S. Hamed
    Citation: Advances in Difference Equations 2021 2021:408
  25. A remarkably large number of hypergeometric (and generalized) functions and a variety of their extensions have been presented and investigated in the literature by many authors. In this paper, we introduce fiv...

    Authors: Jihad Younis, Ashish Verma, Hassen Aydi, Kottakkaran Sooppy Nisar and Habes Alsamir
    Citation: Advances in Difference Equations 2021 2021:407
  26. The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-ca...

    Authors: Sina Etemad, Sotiris K. Ntouyas, Atika Imran, Azhar Hussain, Dumitru Baleanu and Shahram Rezapour
    Citation: Advances in Difference Equations 2021 2021:402
  27. In this manuscript, the existence, uniqueness, and stability of solutions to the multiterm boundary value problem of Caputo fractional differential equations of variable order are established. All results in t...

    Authors: Zoubida Bouazza, Mohammed Said Souid and Hatıra Günerhan
    Citation: Advances in Difference Equations 2021 2021:400
  28. In this paper, we consider a fractional COVID-19 epidemic model with a convex incidence rate. The Atangana–Baleanu fractional operator in the Caputo sense is taken into account. We establish the equilibrium po...

    Authors: Amir Khan, Rahat Zarin, Usa Wannasingha Humphries, Ali Akgül, Anwar Saeed and Taza Gul
    Citation: Advances in Difference Equations 2021 2021:387
  29. This research note’s objective is to elaborate on the study of the unsteady MHD natural convective flow of the Jeffery fluid with the fractional derivative model. The fluid flow phenomenon happens between two ...

    Authors: Maryam Asgir, A. A. Zafar, Abdullah M. Alsharif, Muhammad Bilal Riaz and Muhammad Abbas
    Citation: Advances in Difference Equations 2021 2021:384
  30. Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for som...

    Authors: Iqrar Ansari, Khuram Ali Khan, Ammara Nosheen, Ðilda Pečarić and Josip Pečarić
    Citation: Advances in Difference Equations 2021 2021:374
  31. Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce th...

    Authors: Taekyun Kim and Hye Kyung Kim
    Citation: Advances in Difference Equations 2021 2021:361
  32. In this paper, we discuss a generalization to the Cherednik–Opdam integral operator to an abstract space of Boehmians. We introduce sets of Boehmians and establish delta sequences and certain class of convolut...

    Authors: Shrideh Khalaf Al-Omari, Serkan Araci and Mohammed Al-Smadi
    Citation: Advances in Difference Equations 2021 2021:336