Alexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Solvability of a functional differential equation with internal nonlocal integro-differential condition11534559710.21608/jfca.2024.270323.1068ENMoamen OsamaRadwanDepartment of Mathematics, Faculty of Science, Alexandria University, Egypt.0009-0004-0939-6766Ahmed M. A.El-SayedDepartment of Mathematics, Faculty of Science, Alexandria University, Egypt.0000-0001-7092-7950Hanaa RezqallaEbeadDepartment of Mathematics, Faculty of Science, Alexandria University, Alexandria,Egypt.Journal Article20240214This research delves into the investigation of a nonlocal problem characterized by a delay functional differential equation with parameter under the influence of an internal integro-differential condition. Under appropriate assumptions in place, our central objective is to establish the existence and uniqueness of solutions, a task facilitated by the application of the Schauder fixed point theorem.<br />Additionally, we employ the Hyers-Ulam stability concept to thoroughly analyze the stability properties of the problem when subjected to slight perturbations in order to ensure the credibility of the problem.<br />Furthermore, we conduct a specific investigation into the continuous dependence of the unique solution on various factors, providing insights into how changes in these factors affect the behavior of the solution. <br />The combination of the Hyers-Ulam stability concept and the continuous dependence provides a robust framework for conducting a thorough stability analysis of the problem.<br />The study not only contributes to theoretical understanding but also provides practical insights across the analysis of specific cases and instances.https://jfca.journals.ekb.eg/article_345597_e62e789ef6fe93c6c0a8db6000ad4f2a.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701CHEBYSHEV COMPUTATIONAL ALGORITHM FOR EIGHT ORDER BOUNDARY VALUE PROBLEMS11134724110.21608/jfca.2024.263799.1061ENTAIYEOYEDEPODepartment of Applied Sciences, Federal College of Dental Technology and Therapy, Enugu, Nigeria0000-0001-9063-8806Emmanuel AdewaleAdenipekunDepartment of Statistics,Federal Polytechnic Ede, Osun, Nigeria0009-0007-0725-7617GaniyuAjileyeDepartment of Mathematics and Statistics, Federal University Wukari Taraba,670101 Nigeria.Adewole MukailaAjileyeDepartment of Mathematics, University Ilesa, Osun, Nigeria.Journal Article20240118In this research, we present a computational algorithm designed for solving eighth-order Boundary Value Problems(BVPs) using fourth-kind Chebyshev polynomials as basis functions. The method entails assuming an approximate solution employing fourth-kind shifted Chebyshev polynomials. Subsequently, this assumed solution is substituted into the relevant problem. The resulting equation is collocated at evenly spaced points within the interval, resulting in a linear system of equations with unknown Chebyshev coefficient constants. To solve this system, we employ a matrix inversion approach to determine the unknown constants, which are then substituted back into the assumed solution to obtain the desired approximate solution. To validate the effectiveness of the proposed technique, three numerical examples are selected from existing literature. The results obtained from our method are compared with those reported in the literature, demonstrating that the proposed algorithm is not only accurate but also efficient in solving BVPs. Tables and figures are employed to present and illustrate the results.https://jfca.journals.ekb.eg/article_347241_e7113268ba4dfeb7bd67d17afa8e9461.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701NUMERICAL APPROXIMATION OF MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATION BY GALERKIN METHOD WITH CHEBYSHEV POLYNOMIAL BASIS12034833010.21608/jfca.2024.266136.1066ENTAIYEOYEDEPODepartment of Applied Sciences, Federal College of Dental Technology and Therapy, Enugu, Nigeria0000-0001-9063-8806Akanbi KareemBelloDepartment of Mathematics, University of Ilorin, Ilorin, Nigeria.Abubakar UsmanJosDepartment of Mathematics, University of IlorinAbdullahi MuhammedAyindeDepartment of Mathematics, University of Abuja, Abuja, Nigeria.Tomiwa FaatihatMohammedDepartment of Mathematics, University of Ilorin, Ilorin, Nigeria.Journal Article20240128The Galerkin method is a numerical technique used to approximate solutions to Partial Differential Equations (PDEs) or integral equations. To this end, this study employed the Galerkin method to address Multi-Order Fractional Differential Equations (MFDEs), utilizing Chebyshev Polynomials as the basis functions. The approach involved assuming an approximate solution using shifted Chebyshev polynomials, which was then substituted into the given problem. Subsequently, boundary conditions were applied. The residual equation, integrated over the interval of interest along with the weight function, resulted in a linear system of equations with unknown Chebyshev coefficient constants. Maple 18 was utilized to determine these unknown constants, which were then substituted back into the assumed solution to obtain the desired approximate solution. To assess the effectiveness of the proposed technique, numerical examples were solved, and the results were compared with existing literature. The comparison showcased that the suggested algorithm is not only accurate but also efficient in multi-order fractional differential equations. Tables and figures were employed to present and illustrate the obtained results.https://jfca.journals.ekb.eg/article_348330_e717bab4833ce1a547072a8bd4ac195d.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701New results on implicit delay fractional differential equations with impulsive integral boundary conditions11534832310.21608/jfca.2024.264868.1065ENKARTHIKEYANPSri Vasavi CollegePoornimaSDepartment of Mathematics
Sri Vasavi College
ErodeJournal Article20240126Fractional calculus is a natural extension of ordinary calculus, where integrals and derivatives are defined for arbitrary real orders. Since the development of fractional calculus in the 17th century, numerous derivatives have been developed, including Riemann-Liouville, Hadamard, Grunwald-Letnikov, Caputo, and others. There are numerous works devoted to various fractional operators because the selection of an appropriate fractional derivative (or integral) depends on the system under consideration.<br />Impulsive differential equations have played an important role in modeling phenomena. Since the last century, some authors have used impulsive differential systems to describe the model, particularly in describing dynamics of populations subject to abrupt changes as well as other phenomena like harvesting, diseases, and so forth<br />In this paper, we analyze the existence results for implicit fractional differential equations with impulsive delay integral boundary conditions. The sufficient conditions are established to prove the existence results by using the fixed point theorems such as Banach contraction principle and Schaefer’s fixed point theorem. An application is illustrated through an example.https://jfca.journals.ekb.eg/article_348323_e768ea5d43bcafa412f8a48594a78f3b.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Finite Integral Involving Incomplete Aleph-functions and Fresnel Integral1934748610.21608/jfca.2024.258033.1053ENDheerandra ShankerSachanMathematics Department, Asst. Professor, St. Mary's PG College, Vidisha, India0000-0002-9400-3223BalramRajakSt. Mary's PG College, Vidisha (MP)FredericAyantCollege Jean L’herminier, Alleedes Nympheas, 83500 La Seyne-sur-Mer,
FRANCEJournal Article20231225Special functions represent a class of mathematical functions that have<br />achieved a distinct and recognized status within the realms of mathematical analysis,<br />functional analysis, geometry, physics, and diverse practical applications. These<br />functions have emerged as notable tools in these disciplines, owing to their unique<br />properties and inherent significance. Over time, they have become firmly established<br />due to their ability to address specific mathematical challenges and contribute valuable<br />insights to various branches of science and engineering. The primary objective of this<br />paper is to establish a thorough definition of comprehensive finite integrals through the<br />incorporation of both the Fresnel integral and incomplete Aleph-functions. By adopting<br />a unified and general approach, these integrals are shown to yield a diverse range<br />of new outcomes, particularly in specific scenarios. To elucidate and underscore the<br />significance of our contributions, we present a detailed exposition of our findings, accompanied<br />by specific corollaries. These corollaries, in turn, are emphasized as special<br />cases derived directly from the fundamental results outlined in our study.https://jfca.journals.ekb.eg/article_347486_fd6b99a899a93f6f9b840df573636cc6.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701LINEAR AND NONLINEAR FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS: APPLICATION OF COLLOCATION METHOD FOR SOLUTION11034798110.21608/jfca.2024.265490.1064ENGANIYUAJILEYEDepartment of Mathematics and Statistics, Federal University Wukari, Taraba StateLydiaAdikuDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.Jonathan T.AutaDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.Ojo OAdurojaDepartment of Mathematics, University of Ilesa, Ilesa, Osun State, Nigeria.TAIYEOYEDEPOFederal College of Dental Technology and Therapy, Enugu, NigeriaJournal Article20240125This work examines the collocation approach used to solve linear and nonlinear Fredholm integro- differential equations numerically. To convert the problem into an algebraic system of equations, standard collocation points are used. The algebraic equations were then solved using the matrix inversion approach. The method's uniqueness was established, and its efficiency, accuracy, and consistency were demonstrated through the solution of numerical problems.<br />This work examines the collocation approach used to solve linear and nonlinear Fredholm integro- differential equations numerically. To convert the problem into an algebraic system of equations, standard collocation points are used. The algebraic equations were then solved using the matrix inversion approach. The method's uniqueness was established, and its efficiency, accuracy, and consistency were demonstrated through the solution of numerical problems.<br />This work examines the collocation approach used to solve linear and nonlinear Fredholm integro- differential equations numerically. To convert the problem into an algebraic system of equations, standard collocation points are used. The algebraic equations were then solved using the matrix inversion approach. The method's uniqueness was established, and its efficiency, accuracy, and consistency were demonstrated through the solution of numerical problems.https://jfca.journals.ekb.eg/article_347981_b23cc17102f4e212262f04658df86105.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Central index oriented growth analysis of composite entire functions from the view point of (α,β,γ)-order1635003410.21608/jfca.2024.263000.1060ENTanmayBiswasResearch Scientist, Rajbari, Rabindrapally, R. N. Tagore Road
P.O. Krishnagar, P.S.-Katwali, Dist-Nadia, PIN- 741101, West Bengal, IndiaChinmayBiswasDepartment of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Dist.- Nadia, PIN-741302, West Bengal, IndiaSudipta KumarPalDepartment of Mathematics, Jangipur College, P.O.-Jangipur, Dist.-Murshidabad, PIN-742213, West Bengal, IndiaJournal Article20240115Complex analysis is a very important branch of Mathematics and lot of works has been done in this field. One of the important part is growth analysis of entire and meromorphic functions. The ratio $\frac{M_{f}(r)}{M_{g}(r)}$ is called the growth of the entire function $f$ with respect to entire function $g$ in terms of maximum modulus functions. Order and type are classical growth indicators. Definitions of order and type of entire and meromorphic functions have been extended several times. Recently, Bela\"{\i}di et al. [1] have extended the previous ideas and have introduced the definitions of $(\alpha ,\beta ,\gamma )$-order of entire and meromorphic functions. The generalized definitions of order of entire function are obtained by some researchers in terms of central index (see [2, 3]). In this paper, we have discussed on central index oriented some growth properties of composite entire functions on the basis of their $(\alpha ,\beta ,\gamma )$-order and $(\alpha ,\beta ,\gamma )$-lower order, and have generalized some previous works in this line.https://jfca.journals.ekb.eg/article_350034_b0384823d335991f6a1f9e01bf34ab8e.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Some remarks on the central index based growth properties of entire function11235152910.21608/jfca.2024.278363.1081ENTanmayBiswasRajbari, Rabindrapally, R. N. Tagore Road, P.O. -Krishnagar, P.S. -Kotwali, Dist.-Nadia, PIN- 741101, West Bengal, India.ChinmayBiswasDepartment of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Dist.- Nadia, PIN-741302, West Bengal, India.SarmilaBhattacharyyaDepartment of Mathematics, Netaji Mahavidyalaya, P.O.- Arambagh, Dist.-Hooghly, PIN-712601, West Bengal, India.Journal Article20240320In complex analysis, several research works have been done using the concepts of different growth indicators of entire functions such as order, lower order etc. In past decades, the study of the growth properties regarding entire function has usually been done using the concepts of maximum modulus and maximum term. On the other hand, Biswas (J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math., 25(3) (2018), pp. 193-201 and Uzbek Math. J., 2018 (2), (2018), pp. 153-160) has initiated to study some growth properties of composite entire functions using the concepts of central index. Considering the idea, here in this paper, we have compared the central index of the composition of two entire functions with their corresponding left and right factors. Mainly, we have established some results related to the growth rates of composite entire functions on the basis of central index using the ideas of (p,q,t)L-th order, (p,q,t)L-th type and (p,q,t)L-th weak type.https://jfca.journals.ekb.eg/article_351529_29ea265ddf901019d6ffb3af80352c89.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Certain Class of Bi-Univalent Functions Associated with Chebyshev Polynomial and q-Difference Operator1935417310.21608/jfca.2024.275040.1074ENBasheer MMunassarAmran University, Yemen000000021473051XAdelaMoustafaFaculty of Science, Mansoura University0000-0002-3911-0990TahaSultanFaculty of Science, Al Azhar UniversityNasser A.EI-SherbenyFaculty of Science, Al Azhar UniversitySamar MMadianHigher Institute of Engineering and
Technology New Damietta, Egypt.0000-0001-7490-9901Journal Article20240306Using Chebyshev polynomials and q-differential operator, we introduce a novel class of bi-univalent functions within the open unit disk. Initial coefficient bounds for this class are established, emphasizing their significance in complex analysis. Analyticity ensures the representation of these functions through convergent power series, offering a robust tool for comprehending their behavior. The condition of univalence guarantees the one-to-one nature of these functions, preventing multiple mappings of the same point. Researchers employ bi-univalent functions to delve into diverse aspects of complex analysis, unraveling their properties and applications across various mathematical contexts. The exploration encompasses the investigation of geometric properties, including Fekete-Szegö inequality and coefficient bounds, unraveling the intricate interplay between analytic and geometric characteristics. In summary, the study of bi-univalent functions contributes to the depth of complex analysis, providing a nuanced understanding of the relationships between analyticity and univalence. This exploration lays the foundation for advancements in mathematical theory and applications.https://jfca.journals.ekb.eg/article_354173_f6835cde77111403fe6c039c7f96b469.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701NON-OSCILLATORY BEHAVIOR OF HIGHER ORDER HILFER FRACTIONAL DIFFERENCE EQUATION1935417410.21608/jfca.2024.273085.1070ENSivakumarArundhathiDepartment of Mathematics,
Periyar University,
Salem-636 011,
Tamilnadu, IndiaVeluMuthulakshmiDepartment of Mathematics,
Periyar University,
Salem, Tamil Nadu,
India0000-0003-0125-5032Journal Article20240304In this paper, we look into the non-oscillatory behavior of the higher order forced fractional difference equation with positive and negative terms. Since the fractional difference and summation feature has considerably demonstrated its efficiency and validity due to its nonlocal nature and memory interpretation, we employ the Hilfer fractional difference operator, which is an extension of the most widely used Riemann-Liouville and Caputo fractional difference operators. Unlike the method that has been used in the literature, our study is based on certain fundamental concepts derived from discrete fractional calculus as well as mathematical inequalities. In order to aid in arriving at the important end result, a Volterra-type summation equation is constructed as a similar representation of our higher order Hilfer fractional difference problem. We were able to come up with new, easier to implement condition that were satisfied by the non-oscillatory solutions to our analyzed Hilfer fractional difference equation. Further, to demonstrate the empirical reliability of the theoretical finding, we lend a numerical example.https://jfca.journals.ekb.eg/article_354174_19722efc8af0254e993595bff97273b3.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701SOME APPLICATIONS OF FRACTIONAL DERIVATIVE AND MITTAG-LEFFLER FUNCTION11435884710.21608/jfca.2024.275299.1075ENArunDigrasDepartment of Mathematics, Government Polytechnic,
Hingoli - 431 513, Maharashtra, IndiaJournal Article20240307There are different approaches to the fractional calculus which, not being all<br />equivalent, have lead to a certain degree of confusion and several misunderstandings<br />in the literature. Probably for this the fractional calculus is in some way the ”black<br />sheep” of the analysis. In spite of the numerous eminent mathematicians who have<br />worked on it, still now the fractional calculus is object of so many prejudices. In these<br />review lectures we essentially consider and develop two different approaches to the<br />fractional calculus in the framework of the real analysis: the continuous one, based<br />integral operators and the discrete one, based on infinite series of finite differences<br />with increments tendig to zero. Both approaches turn out to be useful in treating our<br />generalized diffusion processes in the theory of probability and stochastic processes.We obtain basic properties like coefficient<br />inequality, distortion and covering theorem, radii of starlikeness, convexity and closeto-convexity, extreme points, Hadamard product, and closure theorems for functions<br />belonging to our classhttps://jfca.journals.ekb.eg/article_358847_104f413002681df09fecbb19c5b5f5af.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701Maximum term oriented growth analysis of composite entire functions from the view point of (α,β,γ)-order1836695210.21608/jfca.2024.280977.1093ENTanmayBiswasResearch Scientist, Rajbari, Rabindrapally, R. N. Tagore Road
P.O. Krishnagar, P.S.-Kotwali, Dist.-Nadia, PIN- 741101, West Bengal, India.ChinmayBiswasDepartment of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Dist.- Nadia, PIN-741302, West Bengal, India.SarmilaBhattacharyyaDepartment of Mathematics, Netaji Mahavidyalaya, P.O.- Arambagh, Dist.-Hooghly, PIN-712601, West Bengal, India.Journal Article20240401The Fundamental Theorem of Classical Algebra- \textquotedblleft If $f(z)$ is a polynomial of degree $n$ with real or complex coefficients, then the equation $f(z)=0$ has at least one root \textquotedblright\ is the most renowned value distribution theorem, and consequently every such given polynomial can take any certain value, real or complex. In the value distribution theory, one study how an entire function assumes some values and, on the other hand, what is the influence of taking certain values on a function in some exact approach. Furthermore it deals with various sides of the behavior of entire functions, one of which is the study of their comparative growth. Accordingly, study of comparative growth properties of composite entire functions in terms of their maximum terms are very well known area of research which we attempt in this paper. Here, in this paper, we have discussed maximum terms based some growth properties of composite<br />entire functions with respect to their left or right factor using $(\alpha,\beta ,\gamma )$-order and $(\alpha ,\beta ,\gamma )$-lower order.https://jfca.journals.ekb.eg/article_366952_cdd74b37474da96c24e80254187c64ff.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701SHEHU TRANSFORM ADOMIAN DECOMPOSITION METHOD FOR THE SOLUTION OF SYSTEMS OF INTEGER AND FRACTIONAL ORDER DIFFERENTIAL EQUATIONS11836695310.21608/jfca.2024.281990.1094ENBabatunde MorufuYisaDepartment of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Nigeria.0000-0002-8621-539XAbdul-wahab TundeTiamiyuDepartment of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Nigeria.Journal Article20240407This paper is concerned with the solution of system of nonlinear fractional and integer order ordinary and partial differential equations. To achieve that aim, a method of solution is proposed which is developed from an integral transform and the well-known Adomian decomposition method. The Shehu transform Adomian decomposition method (STADM) proposed leverage on the unique advantage that Shehu transform, unlike Laplace transform, is applicable to both constant and variable coefficient initial value problems. The nonlinearity in all its forms is handled by developing corresponding Adomian polynomials, while the fractional order derivatives are interpreted in Caputo sense. STADM, when applied to the class of fractional order problems considered in the present work reduces the computational volume and time. The proposed method is applied to selected problems from the literature, and in most cases gives the exact solutions. The results of the problems solved are equally presented in 3D graphs for ease of visualization.https://jfca.journals.ekb.eg/article_366953_d8e0c60aea884bb4c29d472ec84bb3ce.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701A class of Multivalent Meromorphic Functions Involving an Integral Operator1636695410.21608/jfca.2024.290678.1104ENZeinab MSalehBasic Science Dept. Higher. Tec. Instit, The tenth of Ramadan, Egypt.0009-0009-1202-7714Adela OthmanMoustafaMath. Dept., Faculty of Science, Mansoura University, Egypt.0000-0002-3911-0990Samar MohamedMadianHigher Institute of Engineering and
Technology New Damietta, Egypt.0000-0001-7490-9901Journal Article20240518In this paper, for analytic and multivalent functions defined in the punched disc U^{∗}={ϑ∈ℂ:0<|ϑ-δ|<1}=U\{δ}, δ be a fixed point in U. We define the new class of multivalent meromorphic Bazilevič functions M_{δ,p}^{m}(α,β,μ,ρ,γ) associated with the new integral operator J_{δ,p}^{m}(μ,α), from which one can obtain many other new operators using the principle of Hadamard product (or convolution) by taking different values of its parameters. Let P_{k}(ρ,p) be the class of functions θ(ϑ) analytic in U satisfying θ(0)=p and ∫₀^{2π}|((ℜ{θ(ϑ)}-ρ)/(p-ρ))|dθ≤kπ, where ϑ=re^{iθ},k≥2 and 0≤ρ4 and Aouf and Seoudy 1, we prove our theorems.https://jfca.journals.ekb.eg/article_366954_b460ac1b3bbe3c9b8b4eccdc7d6bec2e.pdfAlexandria University, Faculty of ScienceJournal of Fractional Calculus and Applications2090-584X15220240701SOME NEW GENERAL SUMMATION FORMULAS CONTIGUOUS TO THE KUMMER’S FIRST SUMMATION THEOREM11036779710.21608/jfca.2024.263032.1059ENAarif HussainBhatJamia Millia Islamia, New Delhi indiaMohd IdrisQureshiJamia Millia Islamia. New Delhi, IndiaJavidMajidJamia Millia Islamia, New Delhi IndiaJournal Article20240116Due to the great success of hypergeometric functions of one<br />variable, a number of hypergeometric functions of two or more variables<br />have been introduced and explored. The aim of this paper is to provide the<br />extensions and generalizations of Kummer’s first summation theorem for<br />the higher-order hypergeometric series, where numeratorial and denominatorial<br />parameters differ by positive integers, in the form of<br />r+2Fr+1[a, b, {nr + ζr} ; 1 + a − b + m, {ζr} ; −1],<br />with suitable convergence conditions. Where ζr is set of complex or real<br />numbers, {nr} is set of positive integers and suitable restrictions on the<br />value of m.<br />1. INTRODUCTION AND PRELIMINARIES<br />The enormous popularity and broad usefulness of the hypergeometric function<br />2F1 and the generalized hypergeometric functions pFq (p, q ∈ N0) of<br />one variable have inspired and stimulated a large number of researchers to<br />introduce and investigate hypergeometric functions of two or more variables<br />(see, e.g., [3, 16, 7, 21]).https://jfca.journals.ekb.eg/article_367797_5858fcf67c4203dd404228bd3466615d.pdf