Convergence Control Parameter Research Articles

Dynamic Analysis of a Uniform Microbeam Resting on a Nonlinear Foundation Considering Its Curvature Subjected to a Mechanical Impact and Electromagnetic Actuation.

This study proposes an investigation into the nonlinear vibration of a simply supported, flexible, uniform microbeam associated with its curvature considering the mechanical impact, the electromagnetic actuation, the nonlinear Winkler-Pasternak foundation, and the longitudinal magnetic field. The governing differential equations and the boundary conditions are modeled within the framework of a Euler-Bernoulli beam considering an element of the length of the beam at rest and using the second-order approximation of the deflected beam and the Galerkin-Bubnov procedure. In this work, we present a novel characterization of the microbeam and a novel method to solve the nonlinear vibration of the microactuator. The resulting equation of this complex problem is studied using the Optimal Homotopy Asymptotic Method, employing some auxiliary functions derived from the terms that appear in the equation of motion. An explicit closed-form analytical solution is proposed, proving that our procedure is a powerful tool for solving a nonlinear problem without the presence of small or large parameters. The presence of some convergence-control parameters assures the rapid convergence of the solutions. These parameters are evaluated using some rigorous mathematical procedures. The present approach is very accurate and easy to implement, even for complicated nonlinear problems. The local stability near the primary resonance is studied.

Collisional breakage population balance equation: An analytical approach

This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM), called accelerated HAM, recently proposed in (Hussain et al., “Semi-analytical methods for solving non-linear differential equations: A review.”, JMAA, 2023), to solve the collisional breakage population balance model, which is an integro-partial differential equation. We compare our findings with those obtained using the Adomian decomposition method, a well-known technique for solving various forms of differential equations. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise value in accelerated HAM. The other objective of the article is to examine the theoretical convergence analysis of the two proposed methods. Additionally, we conduct theoretical research on the error estimates for both the techniques. To validate our schemes, several numerical examples are considered and the numerical simulations demonstrate that the suggested techniques provide accurate estimates for the solution and moments of the collisional breakage equation.

An implicit semi-analytical technique: Development, analysis and applications

Solving differential equations over extended temporal domains is pivotal for many scientific and engineering endeavors. The progression of numerous phenomena in the natural world occurs over prolonged durations, necessitating a comprehensive understanding of their temporal dynamics to facilitate precise prognostications and judicious decision-making. Existing methodologies such as homotopy perturbation, Adomian decomposition, and homotopy analysis are noted for their precision within limited temporal scopes. An optimized decomposition method proposed by Odibat extends this accuracy over a longer term, yet its solutions become untenable over substantial timeframes. This paper proposes a novel semi-analytical method to yield solutions over greater temporal extents. Moreover, this work integrates a convergence control parameter to bolster the proposed method’s accuracy and operational efficiency. A rigorous theoretical convergence analysis is delineated, substantiating the method’s validity. Three illustrative examples are numerically examined and discussed to validate the proposed approaches empirically.

Steady-state triad resonance between a surface gravity wave and two hydroacoustic waves based on the homotopy analysis method

Under water compressibility, resonant triads can occur within the family of acoustic-gravity waves. This work investigates steady-state triad resonance between a surface gravity wave and two hydroacoustic waves. The water-wave equations are solved as a nonlinear boundary value problem using the homotopy analysis method (HAM). Within the HAM framework, a potential singularity resulting from exact triad resonance is avoided by appropriately choosing the auxiliary linear operator. The resonant hydroacoustic wave component, along with the two primary waves (one hydroacoustic wave and one gravity wave), is considered to determine an initial guess for the velocity potential. Additionally, by selecting an optimal “convergence-control parameter,” the steady-state resonance between a surface gravity wave and two hydroacoustic waves is successfully obtained. It is found that steady-state resonant acoustic-gravity waves are ubiquitous under certain circumstances. The two primary wave components and the resonant hydroacoustic wave component take up most of the energy in the steady-state resonant acoustic-gravity wave system. The amplitude of the resonant hydroacoustic wave component is mainly determined by the primary hydroacoustic wave component, and the amplitudes of both hydroacoustic waves are approximately equal in all cases considered. In addition, when the overall amplitude of the wave system increases, both dimensionless angular frequencies decrease, indicating that the nonlinearity of the entire wave system becomes stronger with an increase in the wave system amplitude. The amplitude of the primary hydroacoustic wave has a relatively large effect on the system's nonlinearity. This work will enrich and deepen our understanding of acoustic-gravity waves.

UTLIZATION OF GENERALIZED HEAT FLUX MODEL ON THERMAL TRANSPORT OF POWELL-EYRING MODEL VIA OHAM WITH HEAT GENEARTION ASPECTS

The focus of this work is on the flow of a spinning Powell-Eyring nanofluid in the boundary layer in three dimensions using magnetohydrodynamics (MHD). The research simulates mass transfer processes and heat transfer processes using non-Fick's mass flux theory and non-Fourier heat flux theory, respectively. Heat transport phenomena are analyzed by the integration of non-linear heat generation/absorption and thermal radiation properties. The boundary layer method is employed to solve a system of nonlinear partial differential equations (PDE) in the mathematical formulation. These equations are converted into nonlinear ordinary differential equations (ODES), and then the optimal OHAM with convergence control parameters is used to solve them. The impact of various physical movement circumstances on concentration and temperature profiles is visually represented through the generation of diagrams. This study provides important new understandings for both applied and scholarly research on complex heat transfer and fluid dynamics mechanisms in the context of three-dimensional MHD boundary layer flows of rotating Powell-Eyring nanofluids.

Sparseness Controlled Proportionate RLS Algorithm for Sparse and Non-Sparse Systems

This paper proposes a novel sparse adaptive technique to handle variable sparsity. The algorithm is based on the measure of sparseness. A proportionate matrix distributes the gain factor for all filter taps during each iteration. Each filter coefficient of the adaptive filter is updated by the corresponding diagonal element of the proportionate matrix, and that depends on the degree of sparseness. The classical recursive least square algorithm (RLS) is amended by accommodating the proportionate matrix to propose the sparseness-controlled proportionate recursive least square (SC-PRLS) algorithm. The convergence control parameter is incorporated into the algorithm to achieve faster convergence and better steady-state error. The performance of Mean squared error and stead state error of the proposed algorithm are also compared with standard RLS and proportionate RLS (PRLS). The simulation results indicate that SC-PRLS is more effective than PRLS and traditional RLS. An increase in the degree of sparseness leads to an increase in steady-state error and it can be controlled by convergence control parameter, while the convergence rate remains intact in the SC-PRLS. It performs superior in sparse as well as in non-sparse environment so this algorithm can handle large variations in the sparseness. Keywords: SC-PRLS, Sparseness Controlled Proportionate Adaptive Algorithm, Sparse and Non-Sparse Systems, RLS.

Numerical study of fractional phi-4 equation

<abstract><p>In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter $ \hbar $, the results presented underscored the remarkable efficiency and accuracy of this approach.</p></abstract>

Developing the analytical solution for the nonlinear bioheat transfer equation through homotopy analysis method along with an optimal convergence-control parameter

Developing the analytical solution for the nonlinear bioheat transfer equation through homotopy analysis method along with an optimal convergence-control parameter

Nonlinear forced vibration of a nanobeam resting on Winkler-Pasternak elastic foundation and subjected to a mechanical impact

The nonlinear governing equations of nanobeam taking into consideration its curvature, resting on an elastic Winkler-Pasternak foundation and based on non-local Euler-Bernoulli beam theory is analyzed. The equation of motion and the boundary conditions are modeled within the framework of a simple supported nanobeam which accounts the presence of a mechanical impact and nonlinear von-Karman strain. The resulting nonlinear differential equations are reduced to only one differential equation which is studied by means of the Optimal Auxiliary Functions Method (OAFM). An explicit analytical solution is proposed for a complex problem. The main quality of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution for the initial linear equation and the form of nonlinear term calculated from the above solution of the linear equation. The convergence-control parameters present in the auxiliary functions are evaluated by a rigorous mathematical procedure. The obtained solutions are in very good agreement with the numerical solution.

Nonlinear Vibrations and Stability Analysis of GPL Reinforced Pipes Conveying Fluid Excited by Arbitrary Initial Conditions: An Optimized Analytical Solution

In this study, nonlinear vibrations and stability of multilayer functionally graded (FG) pipes conveying fluid laying on nonlinear elastic foundation and reinforced by graphene nanoplatelets (GPLs) are investigated with an emphasis on optimized analytical method. Various types of GPL distribution patterns are considered and to estimate the mechanical characteristics of the reinforced pipe the Halpin–Tsai micromechanics theory is applied. The nonlinear differential equation of motion is obtained by using Von–Kármán strain relations in conjunction with the Euler–Bernoulli beam theory and applying Hamilton’s principle. The Galerkin technique has been used for discretizing the partial differential equation. The optimized homotopy analysis method (HAM), as a strong analytical method, is utilized to solve the nonlinear differential equation by considering different initial excitations. The convergence-control parameter is taken into account to guarantee the convergence of the analytical solution. In this study, an exact solution based on HAM for the critical flow velocity and [Formula: see text]th nonlinear frequency is presented. Additionally, series solutions for the nonlinear time responses of the transverse and longitudinal vibrations of fluid-conveying pipe based on optimized HAM are obtained for the first time. In the numerical results, the effects of arbitrary initial conditions and different physical characteristics on the nonlinear frequencies and time responses are extensively examined. It is shown that the nonlinear frequencies and critical flow velocity of the pipe depend not only on the initial excitation amplitude, but also on the entire initial excitation function applied on the pipe.

Extension of optimal auxiliary function method to non-linear fifth order lax and Swada-Kotera problem

In this article, a semi-analytical approach known as the optimal auxiliary function method is extended to the approximate solution of non-linear partial differential equations. The fifth order lax and swada-kotera equations are taken as test examples. Utilizing the well-known least squares method, the optimal convergence control parameter values in the auxiliary function have been determined. The outcomes of the proposed method are contrasted with those of a new iterative approach and a homotopy perturbation method. It has been demonstrated that the suggested method for solving non-linear partial differential equations is straightforward and rapidly convergent. The numerical outcomes demonstrate the effectiveness and reliability of the suggested approach. Additionally, using higher order approximations can increase the suggested method's accuracy.

Homotopy analysis method and its convergence analysis for a nonlinear simultaneous aggregation-fragmentation model

In this article, we focus on addressing the missing convergence analysis of the homotopy analysis method (HAM) for solving pure aggregation and pure fragmentation population balance equations [Kaur et al. (2023), J. Math. Anal. Appl., 512(2), 126166]. This technique is further extended to determine analytical series solutions for a simultaneous aggregation–fragmentation (SAF) population balance equation. The convergence analysis of the extended approach for a SAF equation is performed using the concept of contraction mapping in the Banach space. The HAM method enables us to derive recursive formulas to obtain series solutions, distinguishing it from traditional numerical approaches. One noteworthy advantage of HAM is its capability to solve both linear and nonlinear differential equations without resorting to discretization, while incorporating a convergence control parameter. Given the complex nature of the SAF equation, only a single analytical solution has been available, specifically for a constant aggregation kernel and a binary breakage kernel with a linear selection function. However, our study presents new series solutions for the number density functions, considering the combination of sum and product aggregation kernels with binary breakage kernels and linear/quadratic selection functions. These particular solutions have not been previously documented in the existing literature. To verify the accuracy and efficiency of the proposed approach, the results with the finite volume scheme [Singh et al. (2021), J. Comput. Phys., 435, 110215] for establishing the accuracy and effectiveness of the proposed approach.

Rational approximation for solving Fredholm integro-differential equations by new algorithm

Abstract In this article, we used a novel semi-analytical approach, named the optimal auxiliary function method (OAFM), to solve integro-differential equations (IDEs). The OAFM includes an auxiliary function and convergence control parameters, which expedite the convergence of the method. To apply the proposed method, some assumptions regarding small or large parameters in the problem are necessary. We present numerical outcomes acquired via the OAFM alongside those obtained from other numerical techniques in tables. Furthermore, we demonstrate the efficacy and ease of implementing the proposed method for various IDEs using 2D graphs.

Semi-analytical methods for solving non-linear differential equations: A review

This article develops a new semi-analytical technique based on the homotopy analysis approach for solving linear or non-linear differential equations and the results are compared to the well-known approaches such as the Adomian decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimized decomposition method (ODM). We discuss the decomposition of the non-linear operator to expedite the HAM solution's convergence to its precise values by using the convergence control parameter. The theoretical convergence analysis and the error estimates are studied. Numerical illustrations show that our proposed scheme improves the accuracy of the non-linear problems discussed in the recently published articles [30] and [31] to an excellent extent and also indicate rapid convergence.

Study of heat transfer in MHD viscoelastic fluid of second grade over a stretching porous sheet with electromagnetic effects and nonuniform source/sink

This work is devoted to the study of non-Newtonian fluid of second grade. This model assumes that the flow and heat transfer in MHD viscoelastic fluid is subject to convective boundary conditions in the presence of electromagnetic field and nonuniform source/sink. The flow is considered over a porous sheet. The governing partial differential equations are converted as ordinary differential equations using suitable similarity transformations. The resulting coupled of nonlinear differential equations are successfully solved by means of Optimal Homotopy Asymptotic Method (OHAM). The effects of different physical parameters such as: the second-grade viscoelastic parameter, thermal radiation parameter or Eckert number are analyzed. Several closed form analytical approximate solutions for dimensionless stream function and temperature are presented for some particular cases. Our procedure is based upon an original construction of the solutions using a moderate number of the convergence-control parameters. These parameters lead to very high precision, comparing our approximate solutions with numerical solutions. Let us note that every nonlinear differential equation is reduced to two linear differential equations. The values of the convergence-control parameters are determined using rigorous mathematical procedures. OHAM is an iterative approach which rapidly converges after only one approximation, is effective, explicit, simple and easy to apply. These remarks are in fact the true power of our procedure to solving nonlinear problems without the presence of small or large parameters in the governing equations or boundary/initial conditions.

Investigating nonlinear forced vibration of functionally graded nanobeam based on the nonlocal strain gradient theory considering mechanical impact, electromagnetic actuator, thickness effect and nonlinear foundation

The present study proposes an investigation to nonlinear vibration response of a functionally graded (FG) nanobeam based on nonlocal strain gradient theory (NSGT) taking into consideration the longitudinal magnetic field and the thickness effect. The nonlinear terms from the governing equation are associated with the curvature of the nanobeam, the nonlinear foundation, the mean axial extension of the nanobeam and electromagnetic actuator for pinned-pinned nanobeam and the effect of discontinuities marked by the presence of the Dirac function (mechanical impact).The governing differential equation of motion and the boundary conditions are modeled within the framework of a simple supported Euler-Bernoulli nanobeam which accounts for the nonlinear von-Kármán strain and using so-called Hamilton's principle. To truncate the continuous system having an infinite degree of freedom, the Galerkin-Bubnov procedure is applied. The resulting nonlinear differential equation is studied by Optimal Auxiliary Functions Method. An explicit analytical solution is proposed for a complex problem near the primary resonance. Our proper procedure is a powerful tool to solve a nonlinear problem without presence of any small parameters. The main characteristic of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution formed for initial linear equation and the form of nonlinear terms calculated for the solution of the linear equation. Also, the presence of some convergence-control parameters assures the rapid convergence of the solutions after only a single iteration. These parameters can be evaluated by means of some rigorous mathematical procedures. We have a large freedom to select the auxiliary functions and the number of convergence-control parameters. The proposed approach is proved to be very accurate, simple, and easy to implement for complicated nonlinear problems. The potential energy is analyzed in detail for different values of the parameters. The local stability of motion near the primary resonance is established using homotopy perturbation method.

A homotopy analysis method for forced transverse vibrations of simply supported double-beam systems with a nonlinear inner layer

The present study introduces a novel algorithm based on the homotopy analysis method (HAM) to efficiently solve the equation of motion of simply supported transversely and axially loaded double-beam systems. The original HAM was developed for single partial differential equations (PDEs); the current formulation applies to systems of PDEs. The system of PDEs is derived by modeling two prismatic beams interconnected by a nonlinear inner layer as Euler–Bernoulli beams. We employ the Bubnov–Galerkin technique to turn the PDEs’ system into a system of ordinary differential equations that is further solved with the HAM. The flexibility and straightforwardness of the HAM in computing time-dependent components of the system’s transverse deflection and natural frequencies, in conjunction with the observed fast convergence, offer a robust semi-analytical method for analyzing such systems. Finally, the transverse deflection is built through the modal superposition principle. Thanks to a judicious and high-flexibility selection of initial guesses and convergence control parameters, numerical examples confirm that at most six iterations are needed to achieve convergence, and the results are consistent with the selected benchmark cases.

Analytical method for nonlinear memristive systems

This work is devoted to providing an approximate analytical method to analyze memristor devices. One of basic property of the memristor is pinched hysteresis, considered to be a signature of the existence of memristance. The presence of hysteresis defines the material implementation of its memristive effects. This it its fundamental property, which looks more like a nonlinear anomaly. The memristor technology offers lower heat generation as it utilizes less energy. Optimal auxiliary functions method (OAFM) is implemented to find an approximate solution for the state variable in memristor with a very high accuracy. The presence of the auxiliary functions and some optimal convergence-control parameters Ci assure a fast convergence of the solutions.

Analysis of the Vibro-Impact Nonlinear Damped and Forced Oscillator in the Dynamics of the Electromagnetic Actuation

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a constant current generating a static force and by an actuation generating a fast alternative force. The deformations between the solids in contact are supposed to be elastic and the contact is maintained. In this study, a single degree of freedom nonlinear damped oscillator under a static normal load is considered. An analytical approximate solution of this problem is obtained using the Optimal Auxiliary Functions Method (OAFM). By means of some auxiliary functions and introducing so-called convergence-control parameters, a very accurate approximate solution of the governing equation can be obtained. We need only the first iteration for this technique, applying a rigorous mathematical procedure in finding the optimal values of the convergence-control parameters. Local stability by means of the Routh-Hurwitz criteria and global stability using the Lyapunov function are also studied. It should be emphasized that the amplitude of AC excitation voltage is not considered much lower than bias voltage (in contrast to other studies). Also, the Hertzian contact coupled with EA is analytically studied for the first time in the present work. The approximate analytical solution is determined with a high accuracy on two domains. Local stability is established in five cases with some cases depending on the trace of the Jacobian matrix and of the discriminant of the characteristic equation. In the study of global stability, the estimate parameters which are components of the Lyapunov function are given in a closed form and a graphical form and therefore the Lyapunov function is well-determined.

Analytic Study of Coupled Burgers’ Equation

In this paper, we construct an analytical solution of the coupled Burgers’ equation, using the homotopy analysis method, which is a semi-analytical method, the approximate solution obtained by this method is convergent for different values of the convergence control parameter ℏ, the optimal value of ℏ corresponding with the minimum error to be determined by the residual. The results obtained by the present method are compared with other obtained solutions by different numerical methods.