Convergence Control Parameter Research Articles

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

On fast queue consensus of discrete-time second-order multi-agent networks over directed topologies

Queue consensus, which aims to drive a set of agents to form a prescribed queue while moving with a common reference velocity, has attracted much attention due to its wide applications. This paper investigates the problem of fast queue consensus for a class of directed multi-agent networks with discrete-time second-order agent dynamics. First, a discrete-time consensus-based control protocol is constructed, and the necessary and sufficient condition to achieve queue consensus is presented. Then, the explicit formulas for the control parameters to achieve the fastest convergence are derived. It is found that the convergence rate of the reference velocity dominates the convergence performance of the overall network. To further accelerate the convergence rate, the agents' node memory is introduced to redesign the update rule of the reference velocity. Furthermore, the analytic formulas of the improved optimal convergence rate and control parameters are established. Finally, numerical examples are given to validate the effectiveness of the proposed theoretical results.

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.

The exploration of thin liquid film flow under the influences of various physical parameters by a new analytical innovation

AbstractA new analytical innovation is utilized to explore the thin liquid film flow of magnetohydrodynamic (MHD) fluid over an unsteady porous stretching surface. Here magnetic field, thermal radiation, and variable viscosity are taken. The self‐similarity variables have been used to transform the modeled partial differential equations into a set of non‐linear coupled differential equations. These non‐linear differential equations for the velocity and temperature profiles have been tackled through an innovation containing homotopy, auxiliary functions, and convergence control parameters. This method is called the second configuration of the Optimal Homotopy Asymptotic Method and is denoted by (OHAM‐2). Galerkin's process is chosen for the optimizations of parameters. The achieved coupled equations are also solved numerically (ND‐Solve Method), and the obtained outcomes are compared with the results elaborated by the suggested algorithm. Here the utmost attention is on the rapid convergence of the proposed algorithm and the consequences of various tangible variables on the velocity and temperature profiles. This straightforward algorithm consists of a few steps but gives better outcomes. The technique can be applied to solve partial differential equations and their systems forming in various disciplines. This technique can also solve Integro differential equations. The residuals achieved by the proposed method for the velocity and temperature fields are shown graphically. The errors and residuals are also taught in Table 1 and Table 2. The effects of parameters are inculcated in Table 3 and Table 4. The results are validated with the published work as shown in Table 5. Nomenclature is indicated in the text below.

Approximate analytical solutions for multispecies convection-dispersion transport equation with variable parameters

Multispecies pollutant migration often occurs in polluted groundwater systems. Most of the multispecies problems that have been dealt in the literature assume constant transport parameters, primarily because analytical solutions for varying parameters become a challenge. The present study analytically solves a two-species convection-dispersion transport equation, considering spatially varying dispersion coefficient and seepage velocity, which corresponds to the steady migration in a steady flow domain. Indeed, the methodology can be adopted for other cases, such as the transient migration in a steady flow domain and transient migration in an unsteady flow domain, without any difficulty. Three kinds of homotopy-based methods, namely the homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimal homotopy asymptotic method (OHAM), are used to derive approximate analytical solutions in the form of truncated series. In homotopy analysis method, the convergence-control parameter ℏ plays a key role in the convergence of the series solution. It is observed that for a specific case of this parameter, namely ℏ=−1, the HAM-based solution recovers the HPM-based solution. For the verification of homotopy-based solutions, we utilize the MATLAB routine pdepe, which efficiently solves a class of parabolic PDEs (single/system). An excellent agreement is found between the derived analytical solutions and the numerical solutions for all three methods. Further, a quantitative assessment is carried out for the derived solutions. Also, convergence theorems are proposed for the series solutions obtained using HAM and OHAM.

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.

A New Computational Technique for Analytic Treatment of Time‐Fractional Nonlinear Equations Arising in Magneto‐Acoustic Waves

This paper presents the study of time‐fractional nonlinear fifth‐order Korteweg–de Vries equations by utilizing an adequate novel technique, namely, the q‐homotopy analysis transform method. The fifth‐order Korteweg–de Vries equation has got its importance in the study of magneto‐sound propagation in plasma, capillary gravity water waves, and the motion of long waves under the influence of gravity in shallow water. To justify the effectiveness and pertinence of the contemplated technique, we take a look at three examples of the time‐fractional fifth‐order Korteweg–de Vries equations. The q‐homotopy analysis transform method offers us to modulate the range of convergence of the series solution using ℏ, called the auxiliary parameter or convergence control parameter. The study of the fractional behaviour of the considered equations expresses the originality of the presented work. There is a visible variation in the obtained solutions for different fractional orders and which can lead to different consequences for future work. As a future research direction, readers can use the hybrid methodologies merging with our projected scheme to achieve better consequences. Additionally, to validate the precision and reliability of the proposed method, we organized suitable numerical simulations. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

Analytic Computational Method for Solving Fractional Nonlinear Equations in Magneto-Acoustic Waves

In this article, we employ a useful and intriguing method known as the ARA-homotopy transform approach to explore the fifth-order Korteweg-de Vries equations that are nonlinear and time-fractional. The study of capillary gravity water waves, magneto-sound propagation in plasma, and the motion of long waves under the effect of gravity in shallow water have all been influenced by Korteweg-de Vries equations. We discuss three instances of the fifth-order time-fractional Korteweg-de Vries equations to demonstrate the efficacy and applicability of the proposed method. Utilizing, also known as the auxiliary parameter or convergence control parameter, the ARA-homotopy transform technique which is a combination between ARA transform and the homotopy analysis method, allows us to modify the convergence range of the series solution. The obtained results show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

Combined Homotopy and Galerkin Stability Analysis of Mathieu-Like Equations

We propose a homotopy analysis method in combination with Galerkin projections to obtain transition curves of Mathieu-like equations. While constructing homotopy, we think of convergence-control parameter as a function of embedding parameter and call it a convergence-control function. Homotopy analysis provides a relation between the parameters of the Mathieu equation that also includes free parameters arising from the convergence-control function. We generate extra nonlinear algebraic equations using Galerkin projections and solve numerically for arriving at transition curves. We demonstrate the usefulness of our method in the case of three distinct versions of linear Mathieu equations by carefully choosing nonlinear and auxiliary operators. Since homotopy analysis does not demand smallness of any of the parameters, our approach has a distinct advantage over perturbation methods in determining transition curves covering a large region of the parameter space. The method is applicable to a wide variety of parametrically excited oscillators.

APPLICATION OF HOMOTOPY ANALYSIS METHOD (HAM) TO THE NON-LINEAR KDV EQUATIONS

In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.

New Series Approach Implementation for Solving Fuzzy Fractional Two-Point Boundary Value Problems Applications

In this work, the fuzzy fractional two-point boundary value problems (FFTBVPs) are analyzed and solved using the fuzzy fractional homotopy analysis method (FF-HAM). Fuzzy set theory mixed with Caputo fractional derivative properties is utilized to produce a new formulation of the standard HAM in the fuzzy domain for the persistence of approximation series solutions for fuzzy fractional differential equations with boundary conditions. The FF-HAM provides a suitable way of controlling the convergence of the series solution through the advantage of the convergence control parameter, which plays a pivotal role in solving a wide range of mathematical problems. The convergence analysis algorithm has been described, along with a graphical representation of the FF-HAM of the proposed applications. The method produces high accuracy solutions with simple implementation for solving linear and nonlinear fuzzy fractional boundary value problems associated with physical application. Also, the obtained results are analyzed and compared with those present in the literature to show the efficiency of the FF-HAM.

Dynamics of the Vibro-Impact Nonlinear Damped and Forced Oscillator under the Influence of the Electromagnetic Actuation

The main objective of the present work is to find an approximate analytical solution for the nonlinear differential equation of the vibro-impact oscillator under the influence of the electromagnetic actuation near the primary resonance. The trigger of vibro-impact regime is due to Hertzian contact. The optimal auxiliary functions method (OAFM) is utilized to give an analytical approximate solution of the problem. The influences of static normal load and electromagnetic actuation near the primary resonance are completely studied. The main novelties of the proposed procedure are the presence of some new adequate auxiliary functions, the introduction of the convergence-control parameters, the original construction of the initial and of the first iteration, and the freedom to choose the method for determining the optimal values of the convergence-control parameters. All these led to an explicit and accurate analytical solution, which is another novelty proposed in the paper. This technique is very accurate, simple, effective, and easy to apply using only the first iteration. A second objective was to perform an analysis of stability of the model using the multiple scales method and the eigenvalues of the Jacobian matrix.

Homotopy Analysis Method Analytical Scheme for Developing a Solution to Partial Differential Equations in Fuzzy Environment

Partial differential equations are known to be increasingly important in today’s research, and their solutions are paramount for tackling numerous real-life applications. This article extended the analytical scheme of the homotopy analysis method (HAM) to develop an approximate analytical solution for Fuzzy Partial Differential Equations (FPDEs). The scheme used its powerful tools, the auxiliary function and convergence-control parameter, in the analysis and optimization, which ensures the convergence of the approximate series solution in addition to considering all necessary concepts from fuzzy set theory to provide high precision in the fuzzy environment. Furthermore, the efficiency was shown by applying the proposed scheme to linear and nonlinear cases of Fuzzy Reaction–Diffusion Equation (FRDE) and Fuzzy Wave Equation (FWE).

Fully nonlinear interfacial periodic waves in a two-layer fluid with a rigid upper boundary and their loads on a cylindrical pile

Based on the fully nonlinear potential flow theory, interfacial periodic gravity waves in a two-layer fluid with a rigid upper boundary are investigated quasi-analytically using the homotopy analysis method (HAM), and the corresponding wave loads on a cylindrical pile are estimated by Morison’s empirical formula. Because of the great freedom in the choices of auxiliary linear operators, initial guesses, and convergence-control parameter, convergent series solutions are obtained in the framework of the HAM. For Boussinesq wave loads on the cylinder in one wave period, there exists a phenomenon of peak–valley mismatch between the upper and lower layer inertial loads; the total drag, total inertial and global loads reach their own peaks almost at the same time when the upper layer load equals the lower layer load. As the wave height increases, the dominant load changes from inertial load to drag load. Previous theoretical studies about internal or interfacial periodic wave loads on a cylinder were mainly based on linear or weakly nonlinear wave models, which causes the lack of corresponding theoretical research in the frame of fully nonlinear wave theory. Compared with linear and weakly nonlinear wave models, the fully nonlinear wave model considered in this paper provides more accurate results for the maximal loads on a slender cylindrical pile exerted by Boussinesq interfacial periodic waves, especially for strongly nonlinear cases. All of these should enhance our understanding of internal periodic waves and their loads on a marine riser.

Explicit series solutions for supersonic flat-plate boundary layer flows

This paper describes explicit series solutions for supersonic flat-plate boundary layer flows that are convergent in the whole spatial domain for Mach numbers of up to 50. These series solutions are achieved by means of the homotopy analysis method (HAM), an analytic technique for highly nonlinear problems. Unlike the analytic approximations given by perturbation methods or other approaches, our explicit series solutions are guaranteed to converge with arbitrary physical parameters because of the so-called “convergence-control parameter” in the HAM framework. Explicit analytic expressions for the local surface skin-friction coefficient and the local heat-transfer coefficient of the supersonic boundary layer flow are also derived. These analytical solutions are found to be in perfect agreement with the corresponding numerical results, allowing the effects of physical parameters on supersonic boundary layer flows to be discussed in detail. The explicit series solutions described in this paper provide a benchmark for supersonic flat-plate boundary layer flows with Mach numbers in the range 0.8≤Ma≤50. To the best of our knowledge, no such explicit series solutions for supersonic flat-plate boundary layer flows have previously been reported. To enable relevant applications, a corresponding Mathematica package is provided to enable convenient access to explicit series solutions for supersonic flat-plate boundary layer flows.

Analytical impact of Carreau nanofluid model under the influence of chemical reaction, Soret and Dufour over inclined stretching cylinder

The present study addresses the Carreau nanofluid flow of two phase dimension with appropriate boundary conditions over stretched inclined cylinder investigated. The transport equations of particle depositions related to the field of pollution studies and designing equipment under the influence of chemical processing. The familiar mathematical scheme method of directly defined inverse mapping (MDDIM) and outcome results compared with R-K-Fehlberg-45 fourth order with shooting method is applied to the transformed equations which leads to approximate analytical and numerical solutions. The MAPLE and BVP4C MATLAB package is used for the computation of the fluid flow result analysis. Also, we examined the average squared residual error εf, εθ and εϕ with the CPU time (Sec.) at ℓ = 1 to 40 approximations, we can gain the solution error within the 256.25 s. It means the convergence control parameters are convergence rate is very fast to get the series solution. The different Carreau nanofluid flow pertinent parameters analyzed with graphical outputs are sketched to view the fluid motion significances. The current research outputs are examined with previous declared results under some limiting range and noticed good validation.

The optimal decomposition method (ODM) for nonlinear problems

The herein coined Optimal Decomposition Method (ODM) is presented; an analytic decomposition approach to solve nonlinear problems that circumvents the well-known convergence barrier within nonlinear decomposition methods, e.g., Adomian Decomposition Method (ADM) and Duan –Rach ADM. ODM initially concerns removing the unknown coefficients from the recursion scheme. In addition, it is powered by multiple Convergence-Control Parameters (CCP) through insertion of an ‘artificial’ parameter. The series of CCPs results in the deployment of the entire convergence potential of the decomposition method. It is shown, in a detailed manner, that ADM and Duan-Rach ADM, i.e., a modified approach preventing the appearance of the unknown coefficients in the Adomian recursion scheme for an l-point BVP ( l≥2), can be generalized only as the special cases of ODM. Therefore, ODM, by itself, moves beyond generalization of the aforementioned methods. The CCPs are computed by minimizing the universal system error. For this purpose, a gradient-based contraction search is exploited together with Genetic Algorithm (GA) to facilitate procuring the CCPs in strongly divergent examples involving several CCPs. Indeed, through examples, namely, a generalized Boundary Value Problem (BVP) test case, the Falkner-Skan equation, a two-state optimal control problem, the Burgers’ equation and a strongly nonlinear problem governing electrohydrodynamic flow of a fluid in a circular cylindrical conduit, it is exemplified that CCPs expand the radius of convergence substantially. In the considered examples, ADM and D-R ADM are strictly divergent with the addition of series terms; on the contrary, ODM serves as a successful remedy on these occasions. In practice, it was found that for most of the nonlinearities, minimizing the system error containing CCPs for lower orders of approximation gives rise to a natural minimization of the system error in higher orders of approximation, i.e., natural convergence.

Approximate Analytical Solutions to Nonlinear Oscillations of Horizontally Supported Jeffcott Rotor

The present paper focuses on nonlinear oscillations of a horizontally supported Jeffcott rotor. An approximate solution to the system of governing equations having quadratic and cubic nonlinearities is obtained in two cases of practical interest: simultaneous and internal resonance. The Optimal Auxiliary Functions Method is employed in this study, and each governing differential equation is reduced to two linear differential equations using the so-called auxiliary functions involving a moderate number of convergence-control parameters. Explicit analytical solutions are obtained for the first time in the literature for the considered practical cases. Numerical validations proved the high accuracy of the proposed analytical solutions, which may be used further in the study of stability and in the design process of some highly performant devices.

An analytical solution for nonlinear vibration of functionally graded porous micropipes conveying fluid in damping medium

This paper presents an analytical solution for nonlinear free vibration of functionally graded porous micropipes conveying fluid embedded in viscous damping medium. By applying Von-Kármán's nonlinear strain relations, strain gradient theory and Hamilton's principle, the nonlinear governing equations of motion are derived in the presence of viscous damping medium. The nonlinear governing equations of motion are discretized and converted to a set of nonlinear ordinary differential equations by using Galerkin's method. For the first time, closed-form expressions are obtained for nonlinear frequency, critical fluid velocity and damped time response of the functionally graded porous micropipe conveying fluid by employing the homotopy analysis method. In addition, the convergence control parameter is considered to diminish the error of the damped vibration response. The effects of the geometrical properties of the micropipe and the viscous damping medium on the vibration of the functionally graded porous micropipe conveying fluid are investigated. The results show that the optimum value of the convergence control parameter dramatically decreases the error of the homotopy solution. Also, the vibration period is increased by the increment of the fluid velocity, damping coefficient, porosity volume fraction, power-law exponent and the length of the functionally graded porous micropipe conveying fluid.

Optimal variational iteration method for parametric boundary value problem

<abstract> <p>Mathematical applications in engineering have a long history. One of the most well-known analytical techniques, the optimal variational iteration method (OVIM), is utilized to construct a quick and accurate algorithm for a special fourth-order ordinary initial value problem. Many researchers have discussed the problem involving a parameter <italic>c</italic>. We solve the parametric boundary value problem that can't be addressed using conventional analytical methods for greater values of <italic>c</italic> using a new method and a convergence control parameter <italic>h</italic>. We achieve a convergent solution no matter how huge <italic>c</italic> is. For the approximation of the convergence control parameter <italic>h</italic>, two strategies have been discussed. The advantages of one technique over another have been demonstrated. Optimal variational iteration method can be seen as an effective technique to solve parametric boundary value problem.</p> </abstract>