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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. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is very easy to implement and handles the restrictive variable that may ruin the physical nature of the problem. Design/methodology/approach: This study combines Sawi transform (ST) and the homotopy perturbation method (HPM) to formulate the idea of Sawi homotopy perturbation transform method (SHPTM). First, this study implements ST to handle the recurrence relation and then incorporates HPM to derive the series solutions of this recurrence relation. ST has the advantage in that it does not require any assumptions or hypothesis for the evaluation of series solutions. Findings: This strategy finds the results very accurate and close to the precise solution. The graphical observations and the surface solution demonstrate that SHPTM is a reliable and powerful scheme for finding the approximate solution of heat flow problems. Originality/value: The study presents an original work. This study develops SHPTM for the approximate solution of two- and three-dimensional heat flow problems. The obtained results and graphical representation demonstrate that SHPTM is a very authentic and reliable approach. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order. Design/methodology/approach: Mohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense. Findings: The results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme. Research limitations/implications: This paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution. Practical implications: This study presents the consistency and authenticity of the graphical solution with the exact solutions. Social implications: This study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions. Originality/value: A new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations. [ABSTRACT FROM AUTHOR]

Purpose: The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system. Design/methodology/approach: First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution. Findings: An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system. Originality/value: The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems. Design/methodology/approach: In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution. Findings: The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models. Originality/value: The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus. [ABSTRACT FROM AUTHOR]

Purpose: This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense. Design/methodology/approach: The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers. Findings: The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach. Research limitations/implications: This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers. Practical implications: In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions. Social implications: This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis. Originality/value: The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors' knowledge, no such work has yet been published in the literature. [ABSTRACT FROM AUTHOR]

Access to safe drinking water and improved sanitation is a fundamental human right and basic ingredient of public health. However, one of the major problems faced by developing countries in the twenty-first century is the lack of access to these facilities. Punjab, the most populous province of Pakistan with more than 50% of the country's population, is no exception. Keeping in view its importance, the current study is an effort to investigate important determinants of access to safe drinking water and improved sanitation in Punjab to ensure the provision of these services to the masses. Multiple Indicator Cluster Survey Household data from 2017 to 2018 has been used for analysis. The results of a logistic regression model revealed that household media exposure, education level of household head, household wealth status, and ethnic background of the household head are some of the important determinants of household access to safe drinking water. For household access to improved sanitation, along with these factors, the role of social norms and place of residence are also important. Particularly, the role of social norms is very profound. Findings from the study suggest that efforts should be made to provide readily available media access, household education level needs to be enhanced, policies should be made to raise the living standard of the poorest households, and the social norm for the use of improved sanitation needs to be promoted. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach. Design/methodology/approach: First, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail. Findings: It is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations. Originality/value: Although many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution. Design/methodology/approach: First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed. Findings: A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He's polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly. Originality/value: The proposed approach is original and very effective because this approach is, to the authors' knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest. [ABSTRACT FROM AUTHOR]