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Article
Publication date: 29 March 2022

Maggie So and Atul Teckchandani

A new way for business leaders to access targeted professional help is via fractional service providers. Fractional service providers can provide tremendous advantages, as they…

Abstract

Purpose

A new way for business leaders to access targeted professional help is via fractional service providers. Fractional service providers can provide tremendous advantages, as they are much more closely associated with the company than outsourcing or consulting service providers, while being more cost effective than full-time employees. A fractional service provider that can be of particular benefit to startups and small businesses is a fractional CFO or Controller – who can provide an organization with the skills to perform all of the activities that a finance and accounting department should perform and provide a consistent leadership voice on all finance-related matters.

Design/methodology/approach

This paper first introduces fractional services and discusses how fractional service providers differ from outsourcing, consulting engagements and full-time employment. Then, it presents an explanation of why fractional service providers may be best suited to manage the finance and accounting functions in a small or medium-sized business. Finally, it discusses factors that business leaders should consider and best practices they should use when using fractional services.

Findings

Using a fractional CFO or Controller will provide an increased focus on the company’s financial health and allow the organization to perform (or oversee) all of the activities that a finance and accounting department should be performing. The scope of work a fractional CFO or Controller performs can be easily modified to meet the needs of the firm, Moreover, they require little direct management. As a result, a fractional CFO or Controller can often be a more cost-effective option than hiring for the finance and accounting function on a full-time basis.

Originality/value

In today’s world, organizations are increasingly seeking ways to maintain effectiveness while also being flexible in how human capital is used. This paper discusses one such flexibility: incorporating fractional service providers. The key premise of this paper is that fractional service providers, specifically fractional CFOs or Controllers, can be an extremely effective way for many organizations, especially small businesses and startups, to get more sophisticated help and guidance in finance and accounting-related matters – thereby acting as an excellent bridge between an ineffective finance and accounting function and creating such a function staffed by full-time employees.

Details

Journal of Business Strategy, vol. 44 no. 4
Type: Research Article
ISSN: 0275-6668

Keywords

Article
Publication date: 17 July 2023

Umer Saeed

The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

Abstract

Purpose

The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.

Design/methodology/approach

The authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.

Findings

The authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.

Originality/value

Since psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Details

Engineering Computations, vol. 40 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 2 June 2023

Dhabaleswar Mohapatra and Snehashish Chakraverty

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking…

Abstract

Purpose

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.

Design/methodology/approach

Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.

Findings

LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.

Originality/value

To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

Details

Engineering Computations, vol. 40 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 17 November 2022

Jinxia Jiang, Haojie Zhao and Yan Zhang

This study aims to investigate the two-dimensional magnetohydrodynamic flow and heat transfer of a fractional Maxwell nanofluid between inclined cylinders with variable thickness…

Abstract

Purpose

This study aims to investigate the two-dimensional magnetohydrodynamic flow and heat transfer of a fractional Maxwell nanofluid between inclined cylinders with variable thickness. Considering the cylindrical coordinate system, the constitutive relation of the fractional viscoelastic fluid and the fractional dual-phase-lag (DPL) heat conduction model, the boundary layer governing equations are first formulated and derived.

Design/methodology/approach

The newly developed finite difference scheme combined with the L1 algorithm is used to numerically solve nonlinear fractional differential equations. Furthermore, the effectiveness of the algorithm is verified by a numerical example.

Findings

Based on numerical analysis, the effects of parameters on velocity and temperature are revealed. Specifically, the velocity decreases with the increase of the fractional derivative parameter α owing to memory characteristics. The temperature increase with the increase of fractional derivative parameter ß due to a decrease in thermal resistance. From a physical perspective, the phase lag of the heat flux vector and temperature gradients τq and τT exhibit opposite trends to the temperature. The ratio τT/τq plays an important role in controlling different heat conduction behaviors. Increasing the inclination angle θ, the types and volume fractions of nanoparticles Φ can increase velocity and temperature, respectively.

Originality/value

Fractional Maxwell nanofluid flows from a fixed-thickness pipe to an inclined variable-thickness pipe, and the fractional DPL heat conduction model based on materials is considered, which provides a basis for the safe and efficient transportation of high-viscosity and condensable fluids in industrial production.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 15 February 2023

Xiankang Luo and Muhammad Nadeem

This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe…

Abstract

Purpose

This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe patterns in two-dimensional systems. The significant results of plot distribution show that the proposed approach is highly authentic and reliable for the fractional-order models.

Design/methodology/approach

The Laplace transform residual power series method (ℒT-RPSM) is the combination of Laplace transform (ℒT) and residual power series method (RPSM). The ℒT is examined to minimize the order of fractional order, whereas the RPSM handles the series solution in the form of convergence. The graphical results of the fractional models are represented through the fractional order α.

Findings

The derived results are obtained in a successive series and yield the results toward the exact solution. These successive series confirm the consistency and accuracy of ℒT-RPSM. This study also compares the exact solutions with the graphical solutions to show the performance and authenticity of the visual solutions. The proposed scheme does not require the restriction of variables and produces the numerical results in terms of a series. This strategy is capable to handle the nonlinear terms very easily for the TFNWS model.

Originality/value

This paper presents the original work. This study reveals that ℒT can perform the solution of fractional-order models without any restriction of variables.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 13 September 2022

Mustafa Turkyilmazoglu

The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a…

Abstract

Purpose

The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided.

Design/methodology/approach

In this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models.

Findings

From the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration.

Originality/value

With the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 2016

Wei Meng, Qian Li and Bo Zeng

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its…

Abstract

Purpose

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its properties.

Design/methodology/approach

This disaggregation method includes three main steps. First, by utilizing Gamma function expanded for integer factorial, this paper expands one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, and gives the analytical expression of fractional order reducing generation operator. Then, studies the commutative law and exponential law of fractional order reducing generation operator. Lastly, gives several examples of fractional order reducing generation operator and verifies the commutative law and exponential law of fractional order reducing generation operator.

Findings

The authors pull the analytical expression of fractional order reducing generation operator and verify that fractional order reducing generation operator satisfies commutative law and exponential law.

Practical implications

Expanding the reducing generation operator would help develop grey prediction model with fractional order operators and widen the application fields of grey prediction models.

Originality/value

The analytical expression of fractional order reducing generation operator, properties of commutative law and exponential law for fractional order reducing generation operator are first studied.

Details

Grey Systems: Theory and Application, vol. 6 no. 1
Type: Research Article
ISSN: 2043-9377

Keywords

Open Access
Article
Publication date: 15 December 2020

Tarikul Islam and Armina Akter

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to…

Abstract

Purpose

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.

Design/methodology/approach

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=i=0nai(DξαG/G)i/i=0nbi(DξαG/G)i.

Findings

Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.

Originality/value

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

Article
Publication date: 30 March 2010

Ahmet Yıldırım

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics…

Abstract

Purpose

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

Findings

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

Originality/value

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 12 August 2020

Amjid Ali, Teruya Minamoto, Umer Saeed and Mujeeb Ur Rehman

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Abstract

Purpose

The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative.

Design/methodology/approach

An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems.

Findings

The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples.

Research limitations/implications

The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity.

Originality/value

Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

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