Shazia Sadiq and Mujeeb ur Rehman
In this article, we present the numerical solution of fractional Sturm-Liouville problems by using generalized shifted Chebyshev polynomials.
Abstract
Purpose
In this article, we present the numerical solution of fractional Sturm-Liouville problems by using generalized shifted Chebyshev polynomials.
Design/methodology/approach
We combine right Caputo and left Riemann–Liouville fractional differential operators for the construction of fractional Sturm–Liouville operators. The proposed algorithm is developed using operational integration matrices of generalized shifted Chebyshev polynomials. We introduce a new bound on the coefficients of the shifted. Chebyshev polynomials subsequently employed to establish an upper bound for error in the approximation of a function by shifted Chebyshev polynomials.
Findings
We have solved fractional initial value problems, terminal value problems and Sturm–Liouville problems by plotting graphs and comparing the results. We have presented the comparison of approximated solutions with existing results and exact numerical solutions. The presented numerical problems with satisfactory results show the applicability of the proposed method to produce an approximate solution with accuracy.
Originality/value
The presented method has been applied to a specific class of fractional differential equations, which involve fractional derivatives of a function with respect to some other function. Keeping this in mind, we have modified the classical Chebyshev polynomials so that they involve the same function with respect to which fractional differentiation is performed. This modification is of great help to analyze the newly introduced polynomials from analytical and numerical point of view. We have compared our numerical results with some other numerical methods in the literature and obtained better results.
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Muhammad Ismail, Mujeeb ur Rehman and Umer Saeed
The purpose of this study is to obtain the numerical scheme of finding the numerical solutions of arbitrary order partial differential equations subject to the initial and…
Abstract
Purpose
The purpose of this study is to obtain the numerical scheme of finding the numerical solutions of arbitrary order partial differential equations subject to the initial and boundary conditions.
Design/methodology/approach
The authors present a novel Green-Haar approach for the family of fractional partial differential equations. The method comprises a combination of Haar wavelet method with the Green function. To handle the nonlinear fractional partial differential equations the authors use Picard technique along with Green-Haar method.
Findings
The results for some numerical examples are documented in tabular and graphical form to elaborate on the efficiency and precision of the suggested method. The obtained results by proposed method are compared with the Haar wavelet method. The method is better than the conventional Haar wavelet method, for the tested problems, in terms of accuracy. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis.
Practical implications
The authors present numerical solutions for nonlinear Burger’s partial differential equations and two-term partial differential equations.
Originality/value
Engineers and applied scientists may use the present method for solving fractional models appearing in applications.
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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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Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Abstract
Purpose
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Design/methodology/approach
The aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.
Findings
The upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.
Originality/value
The numerical method is purposed for solving Hadamard-type fractional differential equations.
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Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Abstract
Purpose
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Design/methodology/approach
The numerical method reduces initial value problem into a system of algebraic equations. The method presented here is also applicable to non-linear differential equations. To deal with non-linear equations, a recursive sequence of approximations is developed using quasi-linearization technique.
Findings
The method is tested on several benchmark problems from the literature. Comparison shows the supremacy of proposed method in terms of robust accuracy and swift convergence. Method can work on several similar types of problems.
Originality/value
It has been demonstrated that many physical systems are modelled more accurately by fractional differential equations rather than classical differential equations. Therefore, it is vital to propose some efficient numerical method. The computational technique presented in this paper is based on Newton–Cotes quadrature rule and quasi-linearization. The key feature of the method is that it works efficiently for non-linear problems.
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Umer Saeed, Mujeeb ur Rehman and Qamar Din
The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate…
Abstract
Purpose
The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.
Design/methodology/approach
The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.
Findings
The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.
Originality/value
Many engineers can use the presented method for solving their nonlinear fractional models.
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Binod Guragai, Paul D. Hutchison and M. Theodore Farris
The purpose of this research study is to use a large sample of the US companies and investigate the impact of cash-to-cash cycle’s (C2C) length on company profitability and…
Abstract
The purpose of this research study is to use a large sample of the US companies and investigate the impact of cash-to-cash cycle’s (C2C) length on company profitability and liquidity in present and future periods and also examine whether such impact is dependent upon firm size or industry type. The authors investigate the association between C2C length and return on equity (ROE), as well as liquidity ratios for current and future years using linear regression models. The authors further examine such association for separate industries and explore the effect of size on the primary associations investigated. Consistent with prior literature, this study documents that C2C length is negatively (positively) associated with current profitability (liquidity). The authors also find that there is a significant negative association between C2C length and future profitability extending up to three years, but only for firms in the manufacturing industry. This research study shows that C2C length affects a firm’s current financial performance and managers should view C2C management as an important strategic tool. However, the authors caution that C2C management is not a “one size fits all” strategy and managers in smaller firms should pay close attention to their C2C cycle. The authors also show that firms in manufacturing industry will specifically benefit financially over long-term from C2C management. This article complements existing literature that examines the impact of working capital management on a firm’s financial performance and extends the literature by examining such relationship for different industries and firm sizes. Although the authors include various factors (e.g., firm size, leverage, growth, industry, year, and past performance) in regressions to control for observable differences among firms, there might be other unobservable differences that may have an effect on the results documented.
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Yuvika Gupta and Farheen Mujeeb Khan
The purpose of this study is to comprehend how AI aids marketers in engaging customers and generating value for the company by way of customer engagement (CE). CE is a popular…
Abstract
Purpose
The purpose of this study is to comprehend how AI aids marketers in engaging customers and generating value for the company by way of customer engagement (CE). CE is a popular area of research for scholars and practitioners. One area of research that could have far-reaching ramifications with regard to strengthening CE is artificial intelligence (AI). Consequently, it becomes extremely important to understand how AI is helping the marketer reach customers and create value for the firm via CE.
Design/methodology/approach
A detailed approach using both systematic review and bibliometric analysis was used. It involved identifying key research areas, the most influential authors, studies, journals, countries and organisations. Then, a comprehensive analysis of 50 papers was carried out in the four identified clusters through co-citation analysis. Furthermore, a content analysis of 42 articles for the past six years was also conducted.
Findings
Emerging themes explored through cluster analysis are CE concepts and value creation, social media strategies, big data innovation and significance of AI in tertiary industry. Identified themes for content analysis are CE conceptualisation, CE behaviour in social media, CE role in value co-creation and CE via AI.
Research limitations/implications
CE has emerged as a topic of great interest for marketers in recent years. With the rapid growth of digital media and the spread of social media, firms are now embarking on new online strategies to promote CE (Javornik and Mandelli, 2012). In this review, the authors have thoroughly assessed multiple facets of prior research papers focused on the utilisation of AI in the context of CE. The existing research papers highlighted that AI-powered chatbots and virtual assistants offer real-time interaction capabilities, swiftly addressing inquiries, delivering assistance and navigating customers through their experiences (Cheng and Jiang, 2022; Naqvi et al., 2023). This rapid and responsive engagement serves to enrich the customer’s overall interaction with the business. Consequently, this research can contribute to a comprehensive knowledge of how AI is assisting marketers to reach customers and create value for the firm via CE. This study also sheds light on both the attitudinal and behavioural aspects of CE on social media. While existing CE literature highlights the motivating factors driving engagement, the study underscores the significance of behavioural engagement in enhancing firm performance. It emphasises the need for researchers to understand the intricate dynamics of engagement in the context of hedonic products compared to utilitarian ones (Wongkitrungrueng and Assarut, 2020). CEs on social media assist firms in using their customers as advocates and value co-creators (Prahalad and Ramaswamy, 2004; Sawhney et al., 2005). A few of the CE themes are conceptual in nature; hence, there is an opportunity for scholarly research in CE to examine the ways in which AI-driven platforms can effectively gather customer insights. As per the prior relationship marketing studies, it is evident that building relationships reduces customer uncertainty (Barari et al., 2020). Therefore, by using data analysis, businesses can extract valuable insights into customer preferences and behaviour, equipping them to engage with customers more effectively.
Practical implications
The rapid growth of social media has enabled individuals to articulate their thoughts, opinions and emotions related to a brand, which creates a large amount of data for VCC. Meanwhile, AI has emerged as a radical way of providing value content to users. It expands on a broader concept of how software and algorithms work like human beings. Data collected from customer interactions are a major prerequisite for efficiently using AI for enhancing CE. AI not only reduces error rates but, at the same time, helps human beings in decision-making during complex situations. Owing to built-in algorithms that analyse large amounts of data, companies can inspect areas that require improvement in real time. Time and resources can also be saved by automating tasks contingent on customer responses and insights. AI enables the analysis of customer data to create highly personalised experiences. It can also forecast customer behaviour and trends, helping businesses anticipate needs and preferences. This enables proactive CE strategies, such as targeted offers or timely outreach. Furthermore, AI tools can analyse customer feedback and sentiment across various channels. This feedback can be used to make necessary improvements and address concerns promptly, ultimately fostering stronger customer relationships. AI can facilitate seamless engagement across multiple digital channels, ensuring that customers can interact with a brand through their preferred means, be it social media, email, or chat. Consequently, this research proposes that practitioners and companies can use analysis performed by AI-enabled systems on CEB, which can assist companies in exploring the extent to which each product influences CE. Understanding the importance of these attributes would assist companies in developing more memorable CE features.
Originality/value
This study examines how prominent CE and AI are in academic research on social media by identifying research gaps and future developments. This research provides an overview of CE research and will assist academicians, regulators and policymakers in identifying the important topics that require investigation.