Journal of Fractional Calculus and Applications
https://jfca.journals.ekb.eg/
Journal of Fractional Calculus and Applicationsendaily1Mon, 01 Jul 2024 00:00:00 +0300Mon, 01 Jul 2024 00:00:00 +0300Solvability of a functional differential equation with internal nonlocal integro-differential condition
https://jfca.journals.ekb.eg/article_345597.html
This 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.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.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. 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.The study not only contributes to theoretical understanding but also provides practical insights across the analysis of specific cases and instances.CHEBYSHEV COMPUTATIONAL ALGORITHM FOR EIGHT ORDER BOUNDARY VALUE PROBLEMS
https://jfca.journals.ekb.eg/article_347241.html
In 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.NUMERICAL APPROXIMATION OF MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATION BY GALERKIN METHOD WITH CHEBYSHEV POLYNOMIAL BASIS
https://jfca.journals.ekb.eg/article_348330.html
The 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.New results on implicit delay fractional differential equations with impulsive integral boundary conditions
https://jfca.journals.ekb.eg/article_348323.html
Fractional 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.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 forthIn 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&rsquo;s fixed point theorem. An application is illustrated through an example.Finite Integral Involving Incomplete Aleph-functions and Fresnel Integral
https://jfca.journals.ekb.eg/article_347486.html
Special functions represent a class of mathematical functions that haveachieved a distinct and recognized status within the realms of mathematical analysis,functional analysis, geometry, physics, and diverse practical applications. Thesefunctions have emerged as notable tools in these disciplines, owing to their uniqueproperties and inherent significance. Over time, they have become firmly establisheddue to their ability to address specific mathematical challenges and contribute valuableinsights to various branches of science and engineering. The primary objective of thispaper is to establish a thorough definition of comprehensive finite integrals through theincorporation of both the Fresnel integral and incomplete Aleph-functions. By adoptinga unified and general approach, these integrals are shown to yield a diverse rangeof new outcomes, particularly in specific scenarios. To elucidate and underscore thesignificance of our contributions, we present a detailed exposition of our findings, accompaniedby specific corollaries. These corollaries, in turn, are emphasized as specialcases derived directly from the fundamental results outlined in our study.LINEAR AND NONLINEAR FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS: APPLICATION OF COLLOCATION METHOD FOR SOLUTION
https://jfca.journals.ekb.eg/article_347981.html
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.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.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.Central index oriented growth analysis of composite entire functions from the view point of (α,β,γ)-order
https://jfca.journals.ekb.eg/article_350034.html
Complex 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.Some remarks on the central index based growth properties of entire function
https://jfca.journals.ekb.eg/article_351529.html
In 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.Certain Class of Bi-Univalent Functions Associated with Chebyshev Polynomial and q-Difference Operator
https://jfca.journals.ekb.eg/article_354173.html
Using 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-Szeg&ouml; 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.NON-OSCILLATORY BEHAVIOR OF HIGHER ORDER HILFER FRACTIONAL DIFFERENCE EQUATION
https://jfca.journals.ekb.eg/article_354174.html
In 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.SOME APPLICATIONS OF FRACTIONAL DERIVATIVE AND MITTAG-LEFFLER FUNCTION
https://jfca.journals.ekb.eg/article_358847.html
There are different approaches to the fractional calculus which, not being allequivalent, have lead to a certain degree of confusion and several misunderstandingsin the literature. Probably for this the fractional calculus is in some way the &rdquo;blacksheep&rdquo; of the analysis. In spite of the numerous eminent mathematicians who haveworked on it, still now the fractional calculus is object of so many prejudices. In thesereview lectures we essentially consider and develop two different approaches to thefractional calculus in the framework of the real analysis: the continuous one, basedintegral operators and the discrete one, based on infinite series of finite differenceswith increments tendig to zero. Both approaches turn out to be useful in treating ourgeneralized diffusion processes in the theory of probability and stochastic processes.We obtain basic properties like coefficientinequality, distortion and covering theorem, radii of starlikeness, convexity and closeto-convexity, extreme points, Hadamard product, and closure theorems for functionsbelonging to our classMaximum term oriented growth analysis of composite entire functions from the view point of (α,β,γ)-order
https://jfca.journals.ekb.eg/article_366952.html
The 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 compositeentire functions with respect to their left or right factor using $(\alpha,\beta ,\gamma )$-order and $(\alpha ,\beta ,\gamma )$-lower order.SHEHU TRANSFORM ADOMIAN DECOMPOSITION METHOD FOR THE SOLUTION OF SYSTEMS OF INTEGER AND FRACTIONAL ORDER DIFFERENTIAL EQUATIONS
https://jfca.journals.ekb.eg/article_366953.html
This 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.A class of Multivalent Meromorphic Functions Involving an Integral Operator
https://jfca.journals.ekb.eg/article_366954.html
In this paper, for analytic and multivalent functions defined in the punched disc U^{&lowast;}={&thetasym;&isin;ℂ:0&lt;|&thetasym;-&delta;|&lt;1}=U\{&delta;}, &delta; be a fixed point in U. We define the new class of multivalent meromorphic Bazilevič functions M_{&delta;,p}^{m}(&alpha;,&beta;,&mu;,&rho;,&gamma;) associated with the new integral operator J_{&delta;,p}^{m}(&mu;,&alpha;), 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}(&rho;,p) be the class of functions &theta;(&thetasym;) analytic in U satisfying &theta;(0)=p and &int;₀^{2&pi;}|((&real;{&theta;(&thetasym;)}-&rho;)/(p-&rho;))|d&theta;&le;k&pi;, where &thetasym;=re^{i&theta;},k&ge;2 and 0&le;&rho;4 and Aouf and Seoudy 1, we prove our theorems.SOME NEW GENERAL SUMMATION FORMULAS CONTIGUOUS TO THE KUMMER’S FIRST SUMMATION THEOREM
https://jfca.journals.ekb.eg/article_367797.html
Due to the great success of hypergeometric functions of onevariable, a number of hypergeometric functions of two or more variableshave been introduced and explored. The aim of this paper is to provide theextensions and generalizations of Kummer&rsquo;s first summation theorem forthe higher-order hypergeometric series, where numeratorial and denominatorialparameters differ by positive integers, in the form ofr+2Fr+1[a, b, {nr + &zeta;r} ; 1 + a &minus; b + m, {&zeta;r} ; &minus;1],with suitable convergence conditions. Where &zeta;r is set of complex or realnumbers, {nr} is set of positive integers and suitable restrictions on thevalue of m.1. INTRODUCTION AND PRELIMINARIESThe enormous popularity and broad usefulness of the hypergeometric function2F1 and the generalized hypergeometric functions pFq (p, q &isin; N0) ofone variable have inspired and stimulated a large number of researchers tointroduce and investigate hypergeometric functions of two or more variables(see, e.g., [3, 16, 7, 21]).