The extension of intervalâvalued and realâvalued functions known as fuzzy intervalâvalued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance âŠ
The extension of intervalâvalued and realâvalued functions known as fuzzy intervalâvalued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance of h âGodunovaâLevin fuzzy convex and preinvex functions and also develop the new generation of the HermiteâHadamard and trapezoidâtype fuzzy fractional integral by the implementation of generalized fuzzy fractional operators having modified version of the BesselâMaitland ( E 1 v BMF) function as its kernel. Moreover, we extract some wellâknown inequalities from our main results.
Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is âŠ
Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is essential to establish the existence and uniqueness of fractional methods. Additionally, convexity theory serves as a fundamental component in the study of fractional integral inequalities due to its defining characteristics and properties. Moreover, there is a strong interconnection between convexity and symmetric theories, allowing results from one to be effectively applied to the other. This correlation has become particularly evident in recent decades, further enhancing their importance in mathematical research. This article investigates the Hermite-Hadamard inequalities and their refinements by implementation of generalized fractional operators through the $s$-convex functions, which are considered in both single and double differentiable forms. The study aims to extend and refine existing inequalities with fractional operator having extended Bessel-Maitland functions as a kernel, providing a more generalized framework. By incorporating these special functions, the results encompass and improve numerous classical inequalities found in the literature, offering deeper insights and broader applicability in mathematical analysis.
Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is âŠ
Fractional integral inequalities play a significant role in both pure and applied mathematics, contributing to the advancement and extension of various mathematical techniques. An accurate formulation of such inequalities is essential to establish the existence and uniqueness of fractional methods. Additionally, convexity theory serves as a fundamental component in the study of fractional integral inequalities due to its defining characteristics and properties. Moreover, there is a strong interconnection between convexity and symmetric theories, allowing results from one to be effectively applied to the other. This correlation has become particularly evident in recent decades, further enhancing their importance in mathematical research. This article investigates the Hermite-Hadamard inequalities and their refinements by implementation of generalized fractional operators through the $s$-convex functions, which are considered in both single and double differentiable forms. The study aims to extend and refine existing inequalities with fractional operator having extended Bessel-Maitland functions as a kernel, providing a more generalized framework. By incorporating these special functions, the results encompass and improve numerous classical inequalities found in the literature, offering deeper insights and broader applicability in mathematical analysis.
The extension of intervalâvalued and realâvalued functions known as fuzzy intervalâvalued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance âŠ
The extension of intervalâvalued and realâvalued functions known as fuzzy intervalâvalued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance of h âGodunovaâLevin fuzzy convex and preinvex functions and also develop the new generation of the HermiteâHadamard and trapezoidâtype fuzzy fractional integral by the implementation of generalized fuzzy fractional operators having modified version of the BesselâMaitland ( E 1 v BMF) function as its kernel. Moreover, we extract some wellâknown inequalities from our main results.
In this paper, we introduce the h-convex concept for interval-valued functions. By using the h-convex concept, we present new Jensen and HermiteâHadamard type inequalities for interval-valued functions. Our inequalities generalize âŠ
In this paper, we introduce the h-convex concept for interval-valued functions. By using the h-convex concept, we present new Jensen and HermiteâHadamard type inequalities for interval-valued functions. Our inequalities generalize some known results.
In this study, we define new classes of convexity called h-GodunovaâLevin and h-GodunovaâLevin preinvexity, through which some new inequalities of HermiteâHadamard type are established. These new classes are the generalization âŠ
In this study, we define new classes of convexity called h-GodunovaâLevin and h-GodunovaâLevin preinvexity, through which some new inequalities of HermiteâHadamard type are established. These new classes are the generalization of several known convexities including the s-convex, P-function, and GodunovaâLevin. Further, the properties of the h-GodunovaâLevin function are also discussed. Meanwhile, the applications of h-GodunovaâLevin Preinvex function are given.
This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely âŠ
This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang-Gao-Tenreiro Machado-Baleanu and Yang-Abdel-Aty-Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional-exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang-Abdel-Aty-Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang-Abdel-Aty-Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.
This endeavour provides a new instance of understanding the velocity of a particle in Brownian motion, using the Fokker-Plank equation. Our treatment is based on the new mathematical tool invoked âŠ
This endeavour provides a new instance of understanding the velocity of a particle in Brownian motion, using the Fokker-Plank equation. Our treatment is based on the new mathematical tool invoked by Yang-Abdel-Aty-Cattani (YAC) to solve the fractional derivative on Fokker-Plank equation. The HPTM and RPSMs, are used to obtain the solution in terms of Prabhakar function. Numerical examples to extend these ideas and to establish the capability and truthfulness of the proposed method, which makes use of YAC-fractional derivative, is also briefly sketched.
In this paper, we aim to determine some results of the generalized BesselâMaitland function in the field of fractional calculus. Here, some relations of the generalized BesselâMaitland functions and the âŠ
In this paper, we aim to determine some results of the generalized BesselâMaitland function in the field of fractional calculus. Here, some relations of the generalized BesselâMaitland functions and the Mittag-Leffler functions are considered. We develop Saigo and RiemannâLiouville fractional integral operators by using the generalized BesselâMaitland function, and results can be seen in the form of FoxâWright functions. We establish a new operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msubsup><mml:mi mathvariant="script">Z</mml:mi><mml:mrow><mml:mi>Îœ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>Ï</mml:mi><mml:mo>,</mml:mo><mml:mi>Îł</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ÎŒ</mml:mi><mml:mo>,</mml:mo><mml:mi>Ο</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>Ï</mml:mi></mml:mrow></mml:msubsup><mml:mi>Ï</mml:mi></mml:math> and its inverse operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mi>Îœ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>,</mml:mo><mml:mi>Ï</mml:mi><mml:mo>,</mml:mo><mml:mi>Îł</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>ÎŒ</mml:mi><mml:mo>,</mml:mo><mml:mi>Ο</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>Ï</mml:mi></mml:mrow></mml:msubsup><mml:mi>Ï</mml:mi></mml:math>, involving the generalized BesselâMaitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the RiemannâLiouville operator and the integral transform (Laplace) of the new operator have been developed.
Abstract In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for $(s+1)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:math> -convex functions involving the $(k,s)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:math> -RiemannâLiouville integrals. Moreover, âŠ
Abstract In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for $(s+1)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:math> -convex functions involving the $(k,s)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:math> -RiemannâLiouville integrals. Moreover, we develop such integral inequalities for h -convex functions involving the k -conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical HermiteâHadamard and trapezoid inequalities.
Abstract In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The âŠ
Abstract In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order FredholmâVolterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authorsâ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.
In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, âŠ
In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, Laplace transform, Mellin transforms, and some relations of extension of extended generalized multi-index Bessel function (E<sup>1</sup>GMBF) with the Laguerre polynomial and Whittaker functions. Further, we also discuss the composition of the generalized fractional integral operator having Appell function as a kernel with the extension of extended generalized multi-index Bessel function and establish these results in terms of Wright functions.
The main goal of this paper is to describe the new version of extended BesselâMaitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we âŠ
The main goal of this paper is to describe the new version of extended BesselâMaitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and RiemannâLiouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized FoxâWright function are presented. Additionally, we discuss a few exceptional cases of the main result.
Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. âŠ
Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs and investigate the existence of fuzzy fractional integral operators (FFIOs) having a generalized BesselâMaitland function as their kernel. Using the class of convexities and pre-invexities FIVFs, we prove some HermiteâHadamard (H-H) and trapezoid-type inequalities by the implementation of FFIOs. Additionally, we obtain other well known inequalities having significant behavior in the field of fuzzy interval analysis.