Q-homotopy Analysis Method Research Articles

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

New modifications of natural transform iterative method and q-homotopy analysis method applied to fractional order KDV-Burger and Sawada–Kotera equations

This manuscript presents enhanced versions of two methods: the natural transform iterative method (NTIM) and the q-homotopy analysis method (q-HAM). These methods harness concepts from Fractional Calculus, particularly leveraging the Caputo fractional derivative operator, to successfully manage the complexities of fractional-order systems. To validate their accuracy and efficiency, we applied the proposed techniques to FPDEs like the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations. Our outcomes, which closely resemble the exact solutions, demonstrate how useful NTIM and q-HAM are for solving difficult FPDEs and improving the study of fractional calculus.

Novel approaches for nonlinear Sine-Gordon equations using two efficient techniques

ABSTRACT In this work, we obtained a new functional matrix using Clique-polynomials of complete graphs K n with n vertices and considered a new approach to solving the Sine–Gordon (SG) equation. The clique polynomial method transforms this equation into a system of algebraic equations. The solution will be drawn with the help of Newton Raphson’s method. Also, we employed the q-homotopy analysis transform method (q-HATM), which is the proper collision of the Laplace transform and the q-homotopy analysis method (q-HAM). To witness the reliability and accuracy of the considered schemes, some illustrations of the SG equation and double SG equation are considered. Here, the SG equation is solved easily and elegantly without using discretization or transformation of the equation by using the q-HATM. Also, in q-HATM, the presence of homotopy and axillary parameters allows us to have a large convergence region. The 3D surfaces of acquired solutions are drawn effectively. The tables of error analysis demonstrate the success of these methods.

The Analytical Improvement of q-Homotopy Analysis Method for Magneto-hydrodynamic (MHD) Jeffrey Hamel Nanofluid Flow

The Analytical Improvement of q-Homotopy Analysis Method for Magneto-hydrodynamic (MHD) Jeffrey Hamel Nanofluid Flow

A homotopy-based computational scheme for two-dimensional fractional cable equation

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper, two types of fractional nonlinear equations in Caputo sense, time-fractional Newell–Whitehead equation (FNWE) and time-fractional generalized Hirota–Satsuma coupled KdV system (HS-cKdVS), are investigated by means of the q-homotopy analysis method (q-HAM). The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just a few terms of the series solution are required in order to obtain better approximation. For the sake of visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.

A new forecasting behavior of fractional model of atmospheric dynamics of carbon dioxide gas

This paper gives insight on the nonlinear mathematical model of fractional order, which defines the dynamic variation of carbon dioxide gas (CO2) concentration in the atmosphere X(t) and examines its solution by applying an efficient technique, namely the q-homotopy analysis generalized transform method (q-HAGTM). This study also highlights the effects of variations in human population N(t) and forest biomass F(t) on the dynamics of CO2 gas concentration in the atmosphere. This technique combines the q-homotopy analysis method (q-HAM) and the generalization of the Laplace transform (GLT) to provide an accurate approximation result. The results prove that the applied technique is suitable for high approximation of the atmospheric carbon dioxide gas concentration fractional model solution and simultaneously proves the efficiency of the applied method. The fluctuating behavior of carbon dioxide gas, forest biomass, and human population concentration is illustrated by the graphical representation of the fractional derivative with variation in time. The convergence and uniqueness of the obtained solutions are evaluated for the studied fractional model. The objective of this work is to assess the pattern of CO2 and its effects on the human population and ecosystem, such as global warming and biomass variation, which in turn affect the patterns of drought, floods, storms, and the spread of climatic and vector-borne diseases.

High-Performance Computational Method for an Extended Three-Coupled Korteweg–de Vries System

This paper calculates numerical solutions of an extended three-coupled Korteweg–de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation.

Unsteady hybrid nanofluid (Cu-UO2/blood) with chemical reaction and non-linear thermal radiation through convective boundaries: An application to bio-medicine

This study is focused on modeling and simulations of hybrid nanofluid flow. Uranium dioxide UO2 nanoparticles are hybrid with copper Cu, copper oxide CuO and aluminum oxide Al2O3 while considering blood as a base fluid. The blood flow is initially modeled considering magnetic effect, non-linear thermal radiation and chemical reactions along with convective boundaries. Then for finding solution of the obtained highly nonlinear coupled system we propose a methodology in which q-homotopy analysis method is hybrid with Galerkin and least square Optimizers. Residual errors are also computed in this study to confirm the validity of results. Analysis reveals that rate of heat transfer in arteries increases up to 13.52 Percent with an increase in volume fraction of Cu while keeping volume fraction of UO2 fixed to 1% in a base fluid (blood). This observation is in excellent agreement with experimental result. Furthermore, comparative graphical study of Cu,CuO and Al2O3 for increasing volume fraction is also performed keeping UO2 volume fraction fixed. Investigation indicates that Cu has the highest rate of heat transfer in blood when compared with CuO and Al2O3. It is also observed that thermal radiation increases the heat transfer rate in the current study. Furthermore, chemical reaction decreases rate of mass transfer in hybrid blood nanoflow. This study will help medical practitioners to minimize the adverse effects of UO2 by introducing hybrid nano particles in blood based fluids.

A Hybrid Analytical Approximate Technique for Solving Two-dimensional Incompressible Flow in Lid-driven Square Cavity Problem

This paper suggests a new technique for finding the analytical approximate solutions to two-dimensional kinetically reduced local Navier-Stokes equations. This new scheme depends combines the q-Homotopy analysis method (q-HAM) , Laplace transform, and Padé approximant method. The power of the new methodology is confirmed by applying it to the flow problem of the lid-driven square cavity. The numerical results obtained by using the proposed method showed that the new technique has good convergence, high accuracy, and efficiency compared with the earlier studies. Moreover, the graphs and tables demonstrate the new approach’s validity.

An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem

The q-homotopy analysis method (q-HAM) in combine with the novel integral transform known as Elzaki transform (ET) leads to an efficient analytical technique called, the q-homotopy analysis Elzaki transform method (q-HAETM). In the present study, the two- dimensional advection–dispersion (AD) problem is investigated using an analytical technique q-HAETM. These equations are mainly used to describe the fate of pollutants in aquifers. The analytical solutions to the AD equations are more interesting since they serve as benchmarks against which numerical solutions can be compared. The novelty of the work is to discuss the two-dimensional (2D) solute transport problem in the fractional sense. The reliability and the efficiency of the considered algorithm are demonstrated by employing the 2D fractional solute transport problem. The solute concentration profile is shown in terms of surface plots. The comparison of the exact solution and the approximate solution is done by the 2D plots. The numerical approximate error solutions are presented for different fractional orders. q-HAETM offers us to modulate the range of convergence of the series solution using ℏ, called auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations in comparison with other existing techniques, the effectiveness and reliability of the considered technique are validated. The obtained findings show that the proposed method is very gratifying and examines the complex challenges that arise in science and innovation.

Computational Analysis of Fractional Liénard's Equation With Exponential Memory

AbstractThe fractional model of Liénard's equations is very useful in the study of oscillating circuits. The main aim of this article is to investigate a fractional extension of Liénard's equation by using a fractional operator with exponential kernel. A user friendly analytical algorithm is suggested to obtain the solutions of fractional model of Liénard's equation. The considered computational technique is a combination of q-homotopy analysis method and an integral transform approach. The outcomes of the investigation presented in graphical and tabular forms, which reveal that the suggested computational scheme is very accurate and useful for handling such type of fractional order nonlinear mathematical models.

Novel approach for nonlinear time-fractional Sharma-Tasso-Olever equation using Elzaki transform

In this article, we demonstrated the study of the time-fractional nonlinear Sharma-Tasso-Olever (STO) equation with different initial conditions. The novel technique, which is the mixture of the q-homotopy analysis method and the new integral transform known as Elzaki transform called, q-homotopy analysis Elzaki transform method (q-HAETM) implemented to find the adequate approximated solution of the considered problems. The wave solutions of the STO equation play a vital role in the nonlinear wave model for coastal and harbor designs. The demonstration of the considered scheme is done by carrying out some examples of time-fractional STO equations with different initial approximations. q-HAETM offers us to modulate the range of convergence of the series solution using , called the auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations, the effectiveness and reliability of the considered technique are validated. The implementation of the new integral transform called the Elzaki transform along with the reliable analytical technique called the q-homotopy analysis method to examine the time-fractional nonlinear STO equation displays the novelty of the presented work. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

Study on generalized fuzzy fractional human liver model with Atangana-Baleanu-Caputo fractional derivative.

This study aims to develop a novel fuzzy fractional model for the human liver that incorporates the ABC fractional differentiability, also known as ABC gH-differentiability, based on the generalized Hukuhara derivative. In addition, a novel fuzzy double parametric q-homotopy analysis method with a generalized transform and ABC gH-differentiability is used to deal with the fuzzy mathematical model and examine its convergence analysis. The stability of the unique equilibrium point for the fuzzy fractional human liver model and the existence of a unique solution in the proposed model are investigated using the Arzela-Ascoli theorem and Schauder's fixed-point theory. Some numerical experiments are conducted to visualize better results and test the proposed method's efficacy. The results of the q-HAShTM employing the presented approaches coincide with most of the clinical data, providing it more precise and superior to the generalized Mittag-Leffler function method.

Solution of generalised fuzzy fractional Kaup–Kupershmidt equation using a robust multi parametric approach and a novel transform

This work considers a fuzzy fractional model of the Kaup–Kupershmidt equation with uncertain initial conditions. The uncertain behaviour of capillary gravity and dispersive waves is studied using a robust double parametric fuzzy approach termed the q-Homotopy analysis method with a generalised novel transform. The proposed method’s accuracy and efficiency have been explained by comparing the observation of fuzziness in the solution with several prior accessible results for the crisp case using the double parametric approach. This methodology provides a viable alternative to existing methods for various fuzzy fractional models.

The q-homotopy analysis method for a solution of the Cahn–Hilliard equation in the presence of advection and reaction terms

In this paper, we provide a solution to the Cahn–Hilliard equation using the q-homotopy analysis method (q-HAM). The q-HAM is a more general, simple and widely used method for solving stiff nonlinear partial differential equations. The Cahn–Hilliard equation is a classical model in material sciences that describe spinodal decomposition and phase separation in two-phase flows. Using the q-HAM, the effect of various parameters of physical interest such as diffusive parameter, thickness parameter, advection and reaction terms on concentration is studied. The comparison of the computed solution with the exact solution is presented for some fixed parameter values to validate the solution obtained using the q-HAM.

An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique

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.

Solution for generalized fuzzy fractional Kortewege-de Varies equation using a robust fuzzy double parametric approach

The nonlinear Kortewege-de Varies (KdV) equation is a functional description for modelling ion-acoustic waves in plasma, long internal waves in a density-stratified ocean, shallow-water waves and acoustic waves on a crystal lattice. This paper focuses on developing and analysing a resilient double parametric analytical approach for the nonlinear fuzzy fractional KdV equation (FFKdVE) under gH-differentiability of Caputo fractional order, namely the q-Homotopy analysis method with the Shehu transform (q-HASTM). A triangular fuzzy number describes the Caputo fractional derivative of order α,0<α≤1, for modelling problem. The fuzzy velocity profiles with crisp and fuzzy conditions at different spatial positions are investigated using a robust double parametric form-based q-HASTM with its convergence analysis. The obtained results are compared with existing works in the literature to confirm the efficacy and effectiveness of the method.

The Analysis of the Fractional-Order Navier-Stokes Equations by a Novel Approach

This article introduces modified semianalytical methods, namely, the Shehu decomposition method and q-homotopy analysis transform method, a combination of decomposition method, the q-homotopy analysis method, and the Shehu transform method to provide an approximate method analytical solution to fractional-order Navier-Stokes equations. Navier-Stokes equations are widely applied as models for spatial effects in biology, ecology, and applied sciences. A good agreement between the exact and obtained solutions shows the accuracy and efficiency of the present techniques. These results reveal that the suggested methods are straightforward and effective for engineering sciences models.

The generalized time-fractional Fornberg–Whitham equation: An analytic approach

This work implements a novel analytical approach known as the q-Homotopy Analysis Shehu Transform Method to the time-fractional Fornberg–Whitham equation in Caputo’s sense. The proposed method combines the idea of the q-Homotopy analysis method with the Shehu transform to enhance the efficacy of the proposed method. To demonstrate the correctness of the proposed method, the convergence analysis of the method has been obtained along with its term approximations. Finally, the obtained numerical solution is compared with the available Laplace Adomian decomposition method (LADM) solution and with the exact answer to test the efficiency of the q-HAShTM through the control parameter ħ and n.