Search Results (16)

The Richards equation is the renowned equation for studying the characteristics of infiltration in unsaturated soil areas, such as porous media. This paper’s primary goal is to demonstrate how the water transport problem behaves in unsaturated soil. In this work, we investigated approximate solutions of the non-linear time-fractional Richards equation (TFRE) with the help of the Yang transform iterative method (YTIM) and the homotopy perturbation transform method (HPTM). The methods used are novel and attractive, successfully combining the Yang transform method, the new iterative method, and the homotopy perturbation method. Both techniques use an iterative process with fewer computations to efficiently provide rapidly convergent series-type solutions. Two cases of the TFRE are examined through the Caputo derivative in order to illustrate the effectiveness of the applied methods. Graphs for various fractional orders are drawn to illustrate the physical behavior of the obtained solutions. Numerical comparisons among the derived approximate solutions and the precise solution are provided in order to demonstrate the efficacy of the YTIM and HPTM. The absolute error obtained using the suggested approaches was contrasted with that obtained using the q-homotopy analysis transform method (q-HATM). The impact of changing non-integer, spatial, and temporal parameters on the behavior of the resulting solution is also illustrated graphically. Mathematica software is used to obtain the approximate series solution and to create graphical displays of several fractional orders. The findings show that the proposed techniques are easy to implement and may be used to investigate sophisticated physical systems controlled by time-fractional non-linear partial differential equations. Full article

The analysis of the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation with various initial conditions has been shown in this work. Finding the appropriate approximate solution of the problems under consideration is carried out by implementing unique strategies that combine the Adomian decomposition method (ADM), and the Generalized integral transform. The proposed method computes the results as a convergent series. The main benefit of the suggested method is that it needs minimal computing effort while producing extremely accurate results. We first apply the fractional Caputo fractional derivative (CFD) and then the Atangana–Baleanu–Caputo (ABC) derivative to solve the fractional STO problem. The nonlinear wave model for harbor and coastal designs heavily relies on the wave solutions of the STO equation. Several cases of time-fractional STO equations with various initial approximations are used to illustrate the schemes under consideration. The efficiency and dependability of the methods under consideration are confirmed by executing suitable numerical simulations. We contrast our findings with those of other approaches, including the Homotopy perturbation method (HPM), and the q-Homotopy analysis Elzaki transform method (q-HAETM). Additionally, the results of using the proposed techniques at different fractional orders are analyzed, showing that their accuracy increases as the value goes from fractional order to integer order. The results gained indicate that the applied scheme is highly satisfying and investigate the complicated nonlinear problems that arise in innovation and science. Full article

The primary objective of this study is to provide a more precise and beneficial mathematical model for assessing corruption dynamics by utilizing non-local derivatives. This research aims to provide solutions that accurately capture the complexities and practical behaviors of corruption. To illustrate how corruption levels within a community change over time, a non-linear deterministic mathematical model has been developed. The authors present a non-integer order model that divides the population into five subgroups: susceptible, exposed, corrupted, recovered, and honest individuals. To study these corruption dynamics, we employ a new method for solving a time-fractional corruption model, which we term the q-homotopy analysis transform approach. This approach produces an effective approximation solution for the investigated equations, and data is shown as 3D plots and graphs, which give a clear physical representation. The stability and existence of the equilibrium points in the considered model are mathematically proven, and we examine the stability of the model and the equilibrium points, clarifying the conditions required for a stable solution. The resulting solutions, given in series form, show rapid convergence and accurately describe the model’s behaviour with minimal error. Furthermore, the solution’s uniqueness and convergence have been demonstrated using fixed-point theory. The proposed technique is better than a numerical approach, as it does not require much computational work, with minimal time consumed, and it removes the requirement for linearization, perturbations, and discretization. In comparison to previous approaches, the proposed technique is a competent tool for examining an analytical outcomes from the projected model, and the methodology used herein for the considered model is proved to be both efficient and reliable, indicating substantial progress in the field. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

This paper introduces an analytical approach for solving the Benney equation using the q-homotopy analysis transform method. The Benney equation is a nonlinear partial differential equation that has applications in diverse areas of physics and engineering. The q-homotopy analysis transform method is a numerical technique that has been successfully employed to solve a broad range of nonlinear problems. By utilizing this method, we derive approximate analytical solutions for the Benney equation. The results demonstrate that this method is a powerful and effective tool for obtaining accurate solutions for the equation. The proposed method offers a valuable contribution to the existing literature on the behavior of the Benney equation and provides researchers with a useful tool for solving this equation in various applications. Full article

In this study, the suggested q-homotopy analysis transform method is used to compute a numerical solution of a fractional parabolic equation, and the solution is obtained in a fast convergent series. The leverage and efficacy of the suggested technique are demonstrated by the test examples provided. The results that were acquired are graphically displayed. The series solution in a sizable admissible domain is handled in an extreme way by the current method. It provides us with a simple means of modifying the solution’s convergence zone. The effectiveness and potential of the suggested algorithm are explicitly shown in the results using graphs. Full article

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotopy analysis generalized transform method is suggested to obtain the solutions of fractional order diffusion equations. The results of this research are demonstrated in graphical and tabular descriptions. This study shows that the applied computational technique is very effective, accurate, and beneficial for managing such kind of fractional order nonlinear models occurring in oil pollution. Full article

In this study, we present a new general solution to a rational epidemiological mathematical model via a recent intelligent method called the natural transform homotopy analysis method (NTHAM), which combines two methods: the natural transform method (NTM) and homotopy analysis method (HAM). To assess the precision and the reliability of the present method, we compared the obtained results with those of the Laplace homotopy perturbation method (LHPM) as well as the q-homotopy analysis Sumudu transform method (q-HASTM), which revealed that the NTHAM is more reliable. The Caputo fractional derivative is employed. It not only gives initial conditions with obvious natural interpretation but is also bounded, meaning that there is no derivative of a constant. The results show that the proposed technique is superior in terms of simplicity, quality, accuracy, and stability and demonstrate the effectiveness of the rational technique under consideration. Full article

In this study, numerical results of a fractional-order multi-dimensional model of the Navier–Stokes equations will be achieved via adoption of two analytical methods, i.e., the Adomian decomposition transform method and the q-Homotopy analysis transform method. The Caputo–Fabrizio operator will be used to define the fractional derivative. The proposed methods will be implemented to provide the series form results of the given models. The series form results of proposed techniques will be validated with the exact results available in the literature. The proposed techniques will be investigated to be efficient, straightforward, and reliable for application to many other scientific and engineering problems. Full article

The existence of man is dependent on nature, and this existence can be disturbed by either man-made devastations or by natural disasters. As a universal phenomenon in nature, symmetry has attracted the attention of scholars. The study of symmetry provides insights into physics, chemistry, biology, and mathematics. One of the most important characteristics in the expressive assessment and development of computational design techniques is symmetry. Yet, mathematical models are an important method of studying real-world systems. The symmetry reflected by such a mathematical model reveals the inherent symmetry of real-world systems. This study focuses on the contagious model of pine wilt disease and symmetry, employing the q-HATM (q-Homotopy Analysis Transform Method) to the leading fractional operator Atangana–Baleanu (AB) to arrive at better understanding. The outgrowths are exhibited in the forms of figures and tables. Finally, the paper helps to analyze the practical theory, assisting the prediction of its manner that corresponds to the guidelines when contemplating the replica. Full article

In the present work, the q-homotopy analysis transform method (q-HATM) was used to generate an analytical solution for the moisture content distribution in a one-dimensional vertical groundwater recharge problem. Three scenarios for the Brooks–Corey model are studied based on linear and nonlinear diffusivity and conductivity functions. The governing nonlinear fractional partial differential equations are solved effectively by the combination of a hybrid analytical technique, which is the combination of the q-homotopy analysis method and the Laplace transform method. Figures and tables are used to discuss the outcomes for fractional values of the time derivative. Mathematica software is used to plot the figures. The examples used in this paper demonstrate the accuracy and competence of the considered algorithm. The acquired results demonstrate the efficiency and reliability of the projected scheme and are also suitable to carry out the highly nonlinear complex problems in a real-world scenario. Full article

The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the $q$-homotopy analysis transform technique ($q$-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order. Full article

This paper presents an efficacious analytical and numerical method for solution of fractional differential equations. This technique, here in named q-HATM (q-homotopy analysis transform method) is applied to a one-dimensional fractional Fornberg–Whitham model and a two-dimensional fractional population model emanating from biological sciences. The overwhelming agreement of our analytical solution by the q-HATM technique with the exact solution indeed establishes the efficacy of q-HATM to solve the fractional Fornberg–Whitham model and the two-dimensional fractional population model. Furthermore, comparisons by means of extensive analysis using numerics, graphs and error analysis are presented to affirm the preference of q-HATM technique over other methods. A variant of the q-HATM using symmetry can also be considered to solve these problems. Full article

This manuscript investigates the fractional Phi-four equation by using $q$ -homotopy analysis transform method ( $q$ -HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and $q$ -homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach’s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations. Full article

The $q$ -homotopy analysis transform method ( $q$ -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology. Full article