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Journal articles on the topic 'Fractional Taylor's theorem'

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Published: 5 June 2025

Last updated: 15 July 2025

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1

GUO, PENG, CHANGPIN LI, and GUANRONG CHEN. "ON THE FRACTIONAL MEAN-VALUE THEOREM." International Journal of Bifurcation and Chaos 22, no. 05 (2012): 1250104. http://dx.doi.org/10.1142/s0218127412501040.

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In this paper, we derive a fractional mean-value theorem both in the sense of Riemann–Liouville and in the sense of Caputo. This new formulation is more general than the generalized Taylor's formula of Kolwankar and the fractional mean-value theorem in the sense of Riemann–Liouville developed by Trujillo.

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2

Chii-Huei, Yu. "Study on Some Properties of Fractional Analytic Function." International Journal of Mechanical and Industrial Technology 10, no. 1 (2022): 31–35. https://doi.org/10.5281/zenodo.7016567.

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<strong>Abstract:</strong> In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional derivative, we study some properties of fractional analytic function, such as fractional Taylor’s theorem, first fractional derivative test, and second fractional derivative test. The major methods used in this paper are fractional Rolle’s theorem, fractional mean value theorem, product rule for fractional derivatives, and a new multiplication of fractional analytic functions. In fact, these new results are generalizations of those results in ordinary calculus. <strong>Keywords:</strong> Jumarie’s modified R-L fractional derivative, fractional analytic function, fractional Taylor’s theorem, first and second fractional derivative test, fractional Rolle’s theorem, fractional mean value theorem, product rule for fractional derivatives, new multiplication. <strong>Title:</strong> Study on Some Properties of Fractional Analytic Function <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Mechanical and Industrial Technology       </strong> <strong>ISSN 2348-7593 (Online)</strong> <strong>Vol. 10, Issue 1, April 2022 - September 2022</strong> <strong>Page No: 31-35</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 23-August-2022</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7016567</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/study-on-some-properties-of-fractional-analytic-function</strong>

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Zine, Houssine, El Mehdi Lotfi, Delfim F. M. Torres, and Noura Yousfi. "Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels." Axioms 11, no. 5 (2022): 231. http://dx.doi.org/10.3390/axioms11050231.

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We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of new relations for nth-weighted generalized fractional integrals and derivatives. As an application, new mean value theorems for generalized weighted fractional operators are obtained. Direct corollaries allow one to obtain the recent Taylor’s and mean value theorems for Caputo–Fabrizio, Atangana–Baleanu–Caputo (ABC) and weighted ABC derivatives.

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4

Uçar, Deniz. "New conformable fractional operator and some related inequalities." Filomat 35, no. 11 (2021): 3597–606. http://dx.doi.org/10.2298/fil2111597u.

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In this study, we introduce a new conformable derivative, namely the beta-conformable derivative. We derive Taylor?s theorem for this derivative. We also investigate some new properties of Taylor?s theorem and some useful related theorems for the beta-conformable derivative. In the light of the new operator, we extend some recent and classical integral inequalities including Steffensen and Hermite-Hadamard inequality.

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Abd El-Salam, F. A. "-Dimensional Fractional Lagrange's Inversion Theorem." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/310679.

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Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.

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Batiha, Iqbal M., Iqbal H. Jebril, Amira Abdelnebi, Zoubir Dahmani, Shawkat Alkhazaleh, and Nidal Anakira. "A New Fractional Representation of the Higher Order Taylor Scheme." Computational and Mathematical Methods 2024 (April 29, 2024): 1–11. http://dx.doi.org/10.1155/2024/2849717.

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In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor’s theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions.

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7

Campos, L. M. B. C. "On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira." International Journal of Mathematics and Mathematical Sciences 13, no. 4 (1990): 687–708. http://dx.doi.org/10.1155/s0161171290000941.

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The classical theorems of Taylor, Lagrange, Laurent and Teixeira, are extended from the representation of a complex functionF(z), to its derivativeF(ν)(z)of complex orderν, understood as either a Liouville (1832) or a Rieman (1847) differintegration (Campos 1984, 1985); these results are distinct from, and alternative to, other extensions of Taylor's series using differintegrations (Osler 1972, Lavoie & Osler & Tremblay 1976). We consider a complex functionF(z), which is analytic (has an isolated singularity) atζ, and expand its derivative of complex orderF(ν)(z), in an ascending (ascending-descending) series of powers of an auxiliary functionf(z), yielding the generalized Teixeira (Lagrange) series, which includes, forf(z)=z−ζ, the generalized Taylor (Laurent) series. The generalized series involve non-integral powers and/or coefficients evaluated by fractional derivatives or integrals, except in the caseν=0, when the classical theorems of Taylor (1715), Lagrange (1770), Laurent (1843) and Teixeira (1900) are regained. As an application, these generalized series can be used to generate special functions with complex parameters (Campos 1986), e.g., the Hermite and Bessel types.

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8

Groza, Ghiocel, and Marilena Jianu. "Functions represented into fractional Taylor series." ITM Web of Conferences 29 (2019): 01017. http://dx.doi.org/10.1051/itmconf/20192901017.

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Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.

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9

Cheng, Jinfa. "On Multivariate Fractional Taylor’s and Cauchy’ Mean Value Theorem." Journal of Mathematical Study 52, no. 1 (2019): 38–52. http://dx.doi.org/10.4208/jms.v52n1.19.04.

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10

Djenina, Noureddine, Adel Ouannas, Taki-Eddine Oussaeif, et al. "On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems." Fractal and Fractional 6, no. 3 (2022): 158. http://dx.doi.org/10.3390/fractalfract6030158.

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This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand.

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11

Rebenda, Josef. "Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders." Symmetry 11, no. 11 (2019): 1390. http://dx.doi.org/10.3390/sym11111390.

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The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system.

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Batiha, Iqbal M., Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh, and Khaled Matarneh. "A Numerical Approach of Handling Fractional Stochastic Differential Equations." Axioms 12, no. 4 (2023): 388. http://dx.doi.org/10.3390/axioms12040388.

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This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

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Wu, Shanhe, Muhammad Adil Khan, and Hidayat Ullah Haleemzai. "Refinements of Majorization Inequality Involving Convex Functions via Taylor’s Theorem with Mean Value form of the Remainder." Mathematics 7, no. 8 (2019): 663. http://dx.doi.org/10.3390/math7080663.

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The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result is used for deriving a new fractional inequality.

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14

Al-Nana, Abeer A., Iqbal M. Batiha, and Shaher Momani. "A Numerical Approach for Dealing with Fractional Boundary Value Problems." Mathematics 11, no. 19 (2023): 4082. http://dx.doi.org/10.3390/math11194082.

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This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order α and the fractional central formula for approximating the Caputo differentiator of order 2α, where 0<α≤1. The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach.

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15

Gaboury, Sébastien, and Richard Tremblay. "An Expansion Theorem Involving H-Function of Several Complex Variables." International Journal of Analysis 2013 (January 30, 2013): 1–6. http://dx.doi.org/10.1155/2013/353547.

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The aim of this present paper is to obtain a general expansion theorem involving H-functions of several complex variables. This is done by making use of a Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives given recently by the authors. Special cases are also computed.

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Nadeem, Muhammad, and Loredana Florentina Iambor. "Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme." Fractal and Fractional 7, no. 7 (2023): 530. http://dx.doi.org/10.3390/fractalfract7070530.

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In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.

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17

Khoshkenar, A., M. Ilie, K. Hosseini, et al. "Further studies on ordinary differential equations involving the $ M $-fractional derivative." AIMS Mathematics 7, no. 6 (2022): 10977–93. http://dx.doi.org/10.3934/math.2022613.

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<abstract><p>In the current paper, the power series based on the $ M $-fractional derivative is formally introduced. More peciesely, the Taylor and Maclaurin expansions are generalized for fractional-order differentiable functions in accordance with the $ M $-fractional derivative. Some new definitions, theorems, and corollaries regarding the power series in the $ M $ sense are presented and formally proved. Several ordinary differential equations (ODEs) involving the $ M $-fractional derivative are solved to examine the validity of the results presented in the current study.</p></abstract>

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18

Gohar, Abdelrahman, Mayada Younes, and Salah Doma. "Gohar Fractional Derivative: Theory and Applications." Journal of Fractional Calculus and Nonlinear Systems 4, no. 1 (2023): 17–34. http://dx.doi.org/10.48185/jfcns.v4i1.753.

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The local fractional derivatives marked the beginning of a new era in fractional calculus. Due to their that have never been observed before in the field, they are able to fill in the gaps left by the nonlocal fractional derivatives and substantially increase the field’s theoretical and applied potential. In this article, we introduce a new local fractional derivative that possesses some classical properties of the integer-order calculus, such as the product rule, the quotient rule, the linearity, and the chain rule. It meets the fractional extensions of Rolle’s theorem and the mean value theorem and has more properties beyond those of previously defined local fractional derivatives. We reveal its geometric interpretation and physical meaning. We prove that a function can be differentiable in its sense without being classically differentiable. Moreover, we apply it to solve the Riccati fractional differential equations to demonstrate that it provides more accurate results with less error in comparison with the previously defined local fractional derivatives when applied to solve fractional differential equations. The numerical results obtained in this work by our local fractional derivative are shown to be in excellent agreement with those produced by other analytical and numerical methods such as the enhanced homotopy perturbation method (EHPM), the improved Adams-Bashforth-Moulton method(IABMM), the modified homotopy perturbation method (MHPM), the Bernstein polynomial method (BPM), the fractional Taylor basis method (FTBM), and the reproducing kernel method (RKM).

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19

Khalouta, Ali, and Abdelouahab Kadem. "New analytical method for solving nonlinear time-fractional reaction-diffusion-convection problems." Revista Colombiana de Matemáticas 54, no. 1 (2020): 1–11. http://dx.doi.org/10.15446/recolma.v54n1.89771.

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In this paper, we propose a new analytical method called generalized Taylor fractional series method (GTFSM) for solving nonlinear timefractional reaction-diffusion-convection initial value problems. Our obtained results are given in the form of a new theorem. The advantage of the proposed method compared with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction. The accuracy and efficiency of the method is tested by means of two numerical examples. Obtained results interpret that the proposed method is very effective and simple for solving different types of nonlinear fractional problems.

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Srivastava, H. M., та Sébastien Gaboury. "New Expansion Formulas for a Family of theλ-Generalized Hurwitz-Lerch Zeta Functions". International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/131067.

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We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.

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Martínez, Francisco, and Mohammed K. A. Kaabar. "On the Martínez–Kaabar Fractal–Fractional Reduced Pukhov Differential Transformation and Its Applications." Mathematics 13, no. 3 (2025): 352. https://doi.org/10.3390/math13030352.

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This paper addresses the extension of the Martinez–Kaabar fractal–fractional calculus (simply expressed as MK calculus) to the context of reduced differential transformation, with applications to the solution of some partial differential equations. Since this differential transformation is derived from the Taylor series expansion of real-valued functions of several variables, it is necessary to develop this theory in the context of such functions. Firstly, classical elements of the analysis of functions of several real variables are introduced, such as the concept of partial derivative and Clairaut’s theorem, in terms of the MK partial α,γ-derivative. Next, we establish the fractal–fractional (FrFr) Taylor formula with Lagrange residue and discuss a sufficient condition for a function of class Cα,γ∞ on an open and bounded set D⊂R2 to be expanded into a convergent infinite series, the so-called FrFr Taylor series. The theoretical study is completed by defining the FrFr reduced differential transformation and establishing its fundamental properties, which will allow the construction of the FrFr reduced Pukhov differential transformation method (FrFrRPDTM). Based on the previous results, this new technique is applied to solve interesting non-integer order linear and non-linear partial differential equations that incorporate the fractal effect. Finally, the results show that the FrFrRPDTM represents a simple instrument that provides a direct, efficient, and effective solution to problems involving this class of partial differential equations.

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Tang, Zhuochao, Zhuojia Fu, HongGuang Sun, and Xiaoting Liu. "An efficient localized collocation solver for anomalous diffusion on surfaces." Fractional Calculus and Applied Analysis 24, no. 3 (2021): 865–94. http://dx.doi.org/10.1515/fca-2021-0037.

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Abstract This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

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Sales, Lázaro Lima de, Jonatas Arizilanio Silva, Eliângela Paulino Bento de Souza, Hidalyn Theodory Clemente Mattos de Souza, Antonio Diego Silva Farias, and Otávio Paulino Lavor. "Gaussian integral by Taylor series and applications." REMAT: Revista Eletrônica da Matemática 7, no. 2 (2021): e3001. http://dx.doi.org/10.35819/remat2021v7i2id4330.

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In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a solution for the Gaussian integral from an unprecedented perspective.

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Prakash, Jay, Sangeeta Singh, and Dr Umesh Sharma. "Investigation of Markovian Process in Server Break-down." INTERANTIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT 08, no. 008 (2024): 1–7. http://dx.doi.org/10.55041/ijsrem37372.

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In day to day life goods produced at a number of places are consumed at different locations and then the problem faced by producer or dealer is to transfer goods from numerous places of origin to various places of consumption such that cost of transportation, time required, damages to the product etc. are optimum. This is called classical transportation problem. In this study we have studied transportation problem solve by Markov Decision Process. Introduction to the operations research is made along with over view of optimization techniques of single objective transportation problem. For literature review in the subject of multi objective fractional as well as linear multi objective transportation problems. In basic concepts of Markov Decision Process are given for further usage in the thesis work. We have stated algorithms for solving different types of multi objective programming problems. In entire work we have used basically linear membership functions, hyperbolic membership function and exponential membership function. These membership functions are Markov Decision Process. In case of linear membership function weighted arithmetic mean, weighted quadratic mean, geometric mean is used to find compromise solution. Almost everywhere we have found that the compromise solutions are fairly close to optimal solution when the problem is solved as single objective function. We have solution procedure for multi objective linear transportation is given using Markov Decision Process. Introduction to fractional programming is made and solutions of multi objective linear fractional programming problems are discussed. We have solutions of multi objective linear fractional transportation problem. In this topic fractional objective functions are approximated as linear functions using Taylor’s theorem and then applied method to solve the multi objective linear transportation problem discussed earlier. We focus on solution to fractional transportation problem with Markov Process coefficients. To multi objective fractional transportation problem with goal programming approach. We deal with general integer programming problem along with fractional transportation problems. The main result of this thesis is a characterization of increasing Marko processes satisfying certain conditions. At the end bibliography and research papers published by authors are given. Keywords: Markov Process, Transportation, Queen analyses Server system.

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Chanchlani, Lata, Subhash Alha, and Jaya Gupta. "Generalization of Taylor’s formula and differential transform method for composite fractional q-derivative." Ramanujan Journal 48, no. 1 (2018): 21–32. http://dx.doi.org/10.1007/s11139-018-9997-7.

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Wang, Jiao. "Polynomials for numerical solutions of space-time fractional differential equations (of the Fokker–Planck type)." Engineering Computations 36, no. 9 (2019): 2996–3015. http://dx.doi.org/10.1108/ec-02-2019-0061.

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Purpose Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type. Design/methodology/approach The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations. Findings Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique. Originality/value Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

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Huang, Li, Xian-Fang Li, Yulin Zhao, and Xiang-Yang Duan. "Approximate solution of fractional integro-differential equations by Taylor expansion method." Computers & Mathematics with Applications 62, no. 3 (2011): 1127–34. http://dx.doi.org/10.1016/j.camwa.2011.03.037.

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Karami-Mollaee, Ali, and Oscar Barambones. "Pitch Control of Wind Turbine Blades Using Fractional Particle Swarm Optimization." Axioms 12, no. 1 (2022): 25. http://dx.doi.org/10.3390/axioms12010025.

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To achieve the maximum power from wind in variable-speed regions of wind turbines (WTs), a suitable control signal should be applied to the pitch angle of the blades. However, the available uncertainty in the modeling of WTs complicates calculations of these signals. To cope with this problem, an optimal controller is suitable, such as particle swarm optimization (PSO). To improve the performance of the controller, fractional order PSO (FPSO) is proposed and implemented. In order to construct this approach for a two-mass WT, we propose a new state feedback, which was first applied to the turbine. The idea behind this state feedback was based on the Taylor series. Then, a linear model with uncertainty was obtained with a new input control signal. Thereafter, the conventional PSO (CPSO) and FPSO were used as optimal controllers for the resulting linear model. Finally, a comparison was performed between CPSO and FPSO and the fuzzy Takagi–Sugeno–Kang (TSK) inference system. The provided comparison demonstrates the advantages of the Taylor series with combination to these controllers. Notably, without the state feedback, CPSO, FPSO, and TSK fuzzy systems cannot stabilize WTs in tracking the desired trajectory.

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Khan, Mohammad Faisal, Suha B. Al-Shaikh, Ahmad A. Abubaker, and Khaled Matarneh. "New Applications of Faber Polynomials and q-Fractional Calculus for a New Subclass of m-Fold Symmetric bi-Close-to-Convex Functions." Axioms 12, no. 6 (2023): 600. http://dx.doi.org/10.3390/axioms12060600.

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Using the concepts of q-fractional calculus operator theory, we first define a (λ,q)-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient bounds for a newly established class using the Faber polynomial expansion method. In addition, the Faber polynomial method is used to examine the Fekete–Szegö problem and the unpredictable behavior of the initial coefficient bounds of the functions that belong to the newly established class of m-fold symmetric bi-close-to-convex functions. Our key results are both novel and consistent with prior research, so we highlight a few of their important corollaries for a comparison.

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Liu, Q. X., J. K. Liu, and Y. M. Chen. "On the theoretical basis of memory-free approaches for fractional differential equations." Engineering Computations 36, no. 4 (2019): 1201–18. http://dx.doi.org/10.1108/ec-08-2018-0389.

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PurposeA nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].Design/methodology/approachMathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.FindingsThough it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.Originality/valueThis study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.

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Di Crescenzo, Antonio, and Alessandra Meoli. "On the fractional probabilistic Taylor's and mean value theorems." Fractional Calculus and Applied Analysis 19, no. 4 (2016). http://dx.doi.org/10.1515/fca-2016-0050.

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Williams, Paul. "Fractional calculus on time scales with Taylor’s theorem." Fractional Calculus and Applied Analysis 15, no. 4 (2012). http://dx.doi.org/10.2478/s13540-012-0043-y.

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AbstractWe present a definition of the Riemann-Liouville fractional calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and fractional calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor’s theorem to incorporate our theory.

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33

Zitane, Hanaa, and Delfim F. M. Torres. "Generalized Taylor’s formula for power fractional derivatives." Boletín de la Sociedad Matemática Mexicana 29, no. 3 (2023). http://dx.doi.org/10.1007/s40590-023-00540-0.

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AbstractWe establish a new generalized Taylor’s formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor’s formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor’s expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.

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34

Zulfeqarr, Fahed, Amit Ujlayan, and Priyanka Ahuja. "A Generalization to Ordinary Derivative and its Associated Integral with some applications." Punjab University Journal of Mathematics, April 25, 2023, 135–48. https://doi.org/10.52280/pujm.2023.550401.

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This paper proposes a generalization to the ordinary derivative, the deformable derivative. For this, we employ a limit approach like the ordinary derivative but use a parameter varying over the unit interval. The definition makes the deformable derivative equivalent to the ordinary derivative because one’s existence implies another. Its intrinsic property of continuously deforming function to its derivative, together with the graphical illustration of linear expression of the function and its derivative, renders sufficient substances to name it deformable derivative. We derive Rolle’s, Mean-value and Taylor’s theorems for the deformable derivative by establishing some of its basic properties. We also define the deformable integral using the fundamental theorem of calculus and discuss associated inverse, linearity, and commutativity property. In addition, we establish a connection between deformable integral and Riemann-Liouville fractional integral. As theoretical applications, we solve some fractional differential equations.

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35

Zahra, Tazeen, Hafiz Muhammad Fahad, and Mujeeb ur Rehman. "On Weighted Fractional Calculus With Respect to Functions." Mathematical Methods in the Applied Sciences, December 16, 2024. https://doi.org/10.1002/mma.10626.

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ABSTRACTThis paper aims to further explore the existing theory of weighted fractional operators with respect to functions. This theory extends some fundamental results of classical Riemann–Liouville and Caputo fractional derivatives to their weighted counterparts involving fractional differentiation and integration with respect to functions. By investigating the fundamental principles of these operators, we establish mean value theorems, Taylor's theorems, and integration by parts formulae. The Leibniz rule is extended for weighted Riemann–Liouville derivatives with respect to functions. Also, we present necessary conditions for the existence and uniqueness of solutions for a class of initial value problems, involving weighted Caputo fractional derivatives with respect to functions, in a Sobolev space.

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Allouch, Nadia, Iqbal Batiha, Iqbal H. Jebril, Shawkat Alkhazaleh, Samira Hamani, and Shaher Momani. "A New Fractional Approach for the Higher-Order q-Taylor Method." Image Analysis and Stereology, November 17, 2024. http://dx.doi.org/10.5566/ias.3286.

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The main goal of this work is to propose a new fractional approach of the higher order q-Taylor method with Initial Value Problems (IVPs) for fractional q-difference equations which is called the Fractional Higher-Order q-Taylor Method (FHOqTM). By applying the generalised q-Taylor theorem, this would be achieved. As a consequence, we calculate the FHOqTM’s local truncation error. Finally, we present numerical applications to validate our results by comparing the exact solution and the approximatesolution resulting from (FHOqTM).

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Fernandez, Arran, and Dumitru Baleanu. "The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag–Leffler kernel." Advances in Difference Equations 2018, no. 1 (2018). http://dx.doi.org/10.1186/s13662-018-1543-9.

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Marzban, Hamid Reza. "An accurate method for fractional optimal control problems governed by nonlinear multi-delay systems." Journal of Vibration and Control, March 13, 2022, 107754632110531. http://dx.doi.org/10.1177/10775463211053182.

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This study introduces a novel framework based on a combination of block-pulse and fractional-order Chebyshev functions. The new framework is a generalization of the fractional-order Chebyshev functions called the fractional hybrid functions. An accurate method is designed for solving nonlinear fractional optimal control problems with fractional multi-delay systems. Two essential linear operators, specifically, the fractional derivative operator and the fractional integral operator are introduced by implementing the Caputo and the Riemann–Liouville fractional operators. The two mentioned operators have a fundamental impact on reducing the computational complexity of the problem under study. Furthermore, these operators enable us to simply transform the principal problem into a new optimization one. Due to the structure of the fractional framework, we can construct an accurate solution for an extensive family of fractional multi-delay systems. By using a generalization of Taylor’s theorem, we prove that the proposed framework is convergent. The reliability, feasibility and accuracy of the new fractional framework are validated through examining a wide range of nontrivial examples.

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Atkinson, Colin, and Adel Osseiran. "Discrete-space time-fractional processes." Fractional Calculus and Applied Analysis 14, no. 2 (2011). http://dx.doi.org/10.2478/s13540-011-0013-9.

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AbstractA time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the diffusion equation generalized by allowing the time derivative to be fractional, i.e. of non-integer order. The properties of the resulting distributions are studied in terms of the Mittag-Leffler function. We discuss the computation of these distribution functions by deriving new global rational approximations for the Mittag-Leffler function that account for both its initial Taylor series and asymptotic power-law tail behaviours. Furthermore, we derive integral representations for both the continuous and the discrete time-fractional distributions and use these to prove a convergence theorem.

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Zafar, Rashida, Mujeeb ur Rehman, and Moniba Shams. "On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series." Advances in Difference Equations 2020, no. 1 (2020). http://dx.doi.org/10.1186/s13662-020-02658-1.

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Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

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Prakash, Amit, and Vijay Verma. "Two efficient computational technique for fractional nonlinear Hirota–Satsuma coupled KdV equations." Engineering Computations ahead-of-print, ahead-of-print (2020). http://dx.doi.org/10.1108/ec-02-2020-0091.

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Purpose The purpose of this paper is to apply an efficient hybrid computational numerical technique, namely, q-homotopy analysis Sumudu transform method (q-HASTM) and residual power series method (RPSM) for finding the analytical solution of the non-linear time-fractional Hirota–Satsuma coupled KdV (HS-cKdV) equations. Design/methodology/approach The proposed technique q-HASTM is the graceful amalgamations of q-homotopy analysis method with Sumudu transform via Caputo fractional derivative, whereas RPSM depend on generalized formula of Taylors series along with residual error function. Findings To illustrate and validate the efficiency of the proposed technique, the authors analyzed the projected non-linear coupled equations in terms of fractional order. Moreover, the physical behavior of the attained solution has been captured in terms of plots and by examining the L2 and L∞ error norm for diverse value of fractional order. Originality/value The authors implemented two technique, q-HASTM and RPSM to obtain the solution of non-linear time-fractional HS-cKdV equations. The obtained results and comparison between q-HASTM and RPSM, shows that the proposed methods provide the solution of non-linear models in form of a convergent series, without using any restrictive assumption. Also, the proposed algorithm is easy to implement and highly efficient to analyze the behavior of non-linear coupled fractional differential equation arisen in various area of science and engineering.

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Keshavarz, Parisa, Tofigh Allahviranloo, Farajollah M. Yaghoobi, and Ali Barahmand. "New Method for Numerical Solution of Z-Fractional Differential Equations." New Mathematics and Natural Computation, December 26, 2020, 1–17. http://dx.doi.org/10.1142/s1793005721500034.

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In this paper, at first, we introduce fractional differential equations with [Formula: see text]-valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler’s method and the probability distribution function. Moreover, the corrected fractional Euler’s method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems’ limitation section of the [Formula: see text]-fractional Initial value problem of order [Formula: see text] with the fuzzy Caputo fractional differential (fractional derivatives are defined on the basis of the Hukuhara differences and the generalized fuzzy derivatives). The probability function is based on exponential distribution function and used to represent the reliability of the problem limitation part. Finally, by two examples, we show that the proposed method can arbitrarily approximate the fractional differential equations with [Formula: see text]-valuation.

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Demir, Ali, Sertaç Erman, Berrak Özgür, and Esra Korkmaz. "Analysis of fractional partial differential equations by Taylor series expansion." Boundary Value Problems 2013, no. 1 (2013). http://dx.doi.org/10.1186/1687-2770-2013-68.

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Almutairi, Bander, Muhammad Kamran, Aamir Farooq, et al. "On the Taylor–Couette flow of fractional oldroyd-B fluids in a cylindrically symmetric configuration using transforms." International Journal of Modern Physics C, January 28, 2022. http://dx.doi.org/10.1142/s0129183122500991.

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The focus in this study is to examine the flow formation of Taylor–Couette (T–C) for some fluids exhibiting non-Newtonian properties in a region of cylindrical annulus due to the effect of imposed stresses on the periphery of the inner cylinder while the outer cylinder is hanging around inert. This tangential shear will be liable for the motion of the fluid through the annulus. It is very often when researchers in different fields like engineering, mathematics and physics come across the complexity that the given mathematical model cannot be solved in the existing space and requires to be transformed in the space in which it can be easily solved. Thus, transforms are being used as a key tool for solving many dominant problems in different fields. Many transformations have been introduced by researchers but for solving problems in this study, we will make use of Hankel transform and Laplace transform to obtain the velocity field and the corresponding shear stresses (SS) for the fractional Oldroyd-B fluid (O-B fluid). In the end, a graphical presentation is given for the comparison of the effect on fluid motion due to different parameters like fractional parameters, relaxation and retardation times.

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