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In this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).
Abstract
Purpose
In this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).
Design/methodology/approach
The proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).
Findings
The author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.
Originality/value
The present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.
Details
Keywords
Muhammad Nadeem, Ji-Huan He and Asad Islam
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…
Abstract
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.
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Keywords
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…
Abstract
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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Tasmia Roshan, Surath Ghosh, Ram P. Chauhan and Sunil Kumar
The fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu…
Abstract
Purpose
The fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu operator which is in Caputo sense. Also, the characteristics of the solutions are described briefly with the help of the advance numerical techniques for the different values of fractional order derivatives. This paper aims to discuss the aforementioned objectives.
Design/methodology/approach
In this work, Adams-Bashforth method and Euler method are used to get the solution of the HIV model. These are the important numerical methods. The comparison results also are described with the physical meaning of the solutions of the model.
Findings
HIV model is analyzed under the view of fractional and AB derivative in Atangana-Baleanu-Caputo sense. The uniqueness of the solution is proved by using Banach Fixed point. The solution is derived with the help of Sumudu transform. Further, the authors employed fractional Adam-Bashforth method and Euler method to enumerate numerical results. The authors have used several values of fractional orders to present the outcomes graphically. The above calculations have been done with the help of MATLAB (R2016a). The numerical scheme used in the proposed study is valid and fruitful, and the same can be used to explore other real issues.
Research limitations/implications
This investigation can be done for the real data sets.
Practical implications
This paper aims to express the solution of the HIV model in a better way with the effect of non-locality, this work is very useful.
Originality/value
In this work, HIV model is developed with the help of Atangana- Baleanu operator in Caputo sense. By using Banach Fixed point, the authors proved that the solution is unique. Also, the solution is presented with the help of Sumudu transform. The behaviors of the solutions are checked for different values of fractional order derivatives with the physical meaning with help of the Adam-Bashforth method and the Euler method.
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Azzh Saad Alshehry, Humaira Yasmin, Rasool Shah, Amjid Ali and Imran Khan
The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation…
Abstract
Purpose
The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.
Design/methodology/approach
In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.
Findings
We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.
Originality/value
Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.
Details
Keywords
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
Keywords
Hong-Yan Liu, Ji-Huan He and Zheng-Biao Li
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable…
Abstract
Purpose
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus.
Design/methodology/approach
This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given.
Findings
Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.
Originality/value
Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.
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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
Keywords
Sumaira Qayyum, Muhammad Ijaz Khan, Tasawar Hayat and Ahmed Alsaedi
The purpose of this study is to analyze the Entropy generation analysis and heat transport in three-dimensional flow between two stretchable disks. Joule heating and heat…
Abstract
Purpose
The purpose of this study is to analyze the Entropy generation analysis and heat transport in three-dimensional flow between two stretchable disks. Joule heating and heat generation/absorption are incorporated in the thermal equation. Thermo-diffusion effect is also considered. Flow is conducting for time-dependent applied magnetic field. Induced magnetic field is not taken into consideration. Velocity and thermal slip conditions at both the disks are implemented. The flow problem is modeled by using Navier–Stokes equations with entropy generation rate and Bejan number.
Design/methodology/approach
Von Karman transformations are used to reduce the nonlinear governing expressions into an ordinary one and then tackled by homotopy analysis method for convergent series solutions. The nonlinear expressions for total entropy generation rate are obtained with appropriate transformations. The impacts of different flow variables on velocity, temperature, entropy generation rate and Bejan number are described graphically. Velocity, temperature and concentration gradients are discussed in the presence of flow variables.
Findings
Axial, radial and tangential velocity profiles show decreasing trend for larger values of velocity slip parameters. For a larger Brinkman number, the entropy generation increases, while a decreasing trend is noticed for Bejan number.
Originality/value
To the best of the authors’ knowledge, no such analyses have been reported in the literature.
Details
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Ismail Onder, Melih Cinar, A. Secer and Mustafa Bayram
The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the…
Abstract
Purpose
The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.
Design/methodology/approach
The main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.
Findings
Thus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.
Originality/value
Illustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.
Details