Perturbation Iteration Method Research Articles

New Analytical and Numerical Solutions for Squeezing Flow between Parallel Plates under Slip

In this article, the effects of physical flow parameters on squeezed fluid between parallel plates are explored through the Darcy porous channel when fluid is moving as a result of the upper plate being squeezed towards the stretchable lower plate, such as velocity slip, thermal slip, solutal slip, thermal stratification parameter, solutal stratification parameter, squeezing number, Darcy number, Prandtl number, and Schmidt number. The governing equations are transformed into a nonlinear ordinary differential equation using the appropriate similarity transformations. The resulting equations are solved by using the perturbation iteration method (PIT) to produce a convergent analytical solution with high accuracy. The phenomena of the squeezing fluid as the plates are moving apart and when they are coming together are illustrated using the resulting analytical solutions. Plots are used to discuss the significant effects of physical parameters on velocity, temperature, and fluid concentration profiles. The skin friction coefficient and Nusselt Sherwood values have graphical interpretations that are listed. For strong velocity slip parameters, the results demonstrate the existence of a minimum velocity profile close to the plate and a growing velocity profile distant from the plate. Additionally, as the slip effects rise, the fluid temperature and concentration both considerably drop. The results of the fourth-order Runge-Kutta method (RK4M) and the presented analytical solutions provided are in excellent agreement.

A new approach in handling one-dimensional time-fractional Schrödinger equations

<abstract> <p>Our aim of this paper was to present the accurate analytical approximate series solutions to the time-fractional Schrödinger equations via the Caputo fractional operator using the Laplace residual power series technique. Furthermore, three important and interesting applications were given, tested, and compared with four well-known methods (Adomian decomposition, homotopy perturbation, homotopy analysis, and variational iteration methods) to show that the proposed technique was simple, accurate, efficient, and applicable. When there was a pattern between the terms of the series, we could obtain the exact solutions; otherwise, we provided the approximate series solutions. Finally, graphical results were presented and analyzed. Mathematica software was used to calculate numerical and symbolic quantities.</p> </abstract>

Analytical Solutions of the Blasius Equation by Perturbation Iteration Method

The Blasius equation is treated by employing the Perturbation Iteration method. Analytical solutions are derived for different perturbation iteration algorithms. Solutions are contrasted with the numerical solution obtained by an adaptive step size Runge-Kutta algorithm. It is found that the series type perturbation iteration solution better represents the behavior inside the boundary layer whereas the exponentially decaying perturbation iteration solution better represents the real solution outside the boundary layer. A composite expansion uniting both solutions and valid over the whole region is constructed using the gamma interval functions. The composite analytical solution is indistinguishable from the numerical one and can replace the numerical solution in calculations.

New Numerical Iteration Schemes Based on Perturbation Iteration Algorithms

Perturbation–Iteration algorithms (PIA) have been developed recently to solve differential equations analytically. A continuous solution in terms of closed form functions as an approximation of the original equation can be found using the method. The method has been implemented to algebraic equations, ordinary and partial differential equations successfully. Based on the formalism developed previously, in this work, purely numerical versions of the perturbation–iteration algorithms are proposed for the first time for first-order nonlinear ordinary differential equations. The new algorithms are called as the Numerical Perturbation–Iteration Algorithms (NPIA) to distinguish them from the continuous analytical ones (PIA) and the method in general will be called the Numerical Perturbation–Iteration Method (NPIM). Iteration schemes based on one correction term in the perturbation solution and first-order derivatives in the Taylor series expansions (NPIA(1,1)), and one correction term in the perturbations and second-order derivatives in the series expansions (NPIA(1,2)) are derived. The numerical results are compared with the exact solutions and a very good match is observed. NPIA(1,2) produced slightly better results compared to NPIA(1,1) with total errors being at least [Formula: see text], [Formula: see text] being the step size for both methods. An improvement suggestion for NPIA(1,1) is proposed in the last section to eliminate unrealistic blow-up solutions which are encountered for some specific equations.

Perturbation Iteration Method Compared with Direct Method and Fuzzy Logic Strategy for Solving An Optimal Control Problem of An Uninfected Hepatitis B Virus Dynamics

This paper aims at solving the optimal control problem of the dynamic of HBV infection under treatment using the perturbation iteration method. This method serves as a tool to determine the approximate solutions of nonlinear equations for which exact solutions cannot be obtained. To test the efficacy of this method, the authors propose to compare the numerical simulation results with those of the direct method and fuzzy logic strategy. The newly used method for solving the above optimal control problem is very important since the findings compared to those obtained from the two other methods are in good agreement with experimental data and they demonstrate the response drugs to the dynamics of uninfected hepatocytes, infected hepatocytes, and free virions for a patient suffering from HBV. Since the perturbation iteration method provides satisfactory results which are close to other used numerical methods, it is an important numerical tool to determine the solution of an optimal control problem. In particular, it provides optimal trajectories in medicine, biology, and other related scientific fields. For instance, the response of treatment as control of the human body ensures the health of patients.

Dynamic behaviors for the graphene nano/microelectromechanical system in a fractal space

The microelectromechanical devices have triggered rocketing interest in various advanced technologies, that is, sensors, material science, and energy harvesting, and now the devices tend to a nanoscale size, making the technology much more attractive in both academic and industrial communities. This paper takes a graphene nano/microelectromechanical system as an example to study its dynamical property, which is the foundation for the optimal design and reliable operation. As the system is inextricably complex with singularity and zero conditions in a fractal space, the iteration perturbation method is used to obtain the periodic solution of the system. The results show clearly the low-frequency property of the system and pull-in instability; furthermore, the fractal dimension affects greatly the system’s dynamical property.

New optimum solutions of nonlinear fractional acoustic wave equations via optimal homotopy asymptotic method-2 (OHAM-2)

The second iteration of the optimal homotopy asymptotic technique (OHAM-2) has been protracted to fractional order partial differential equations in this work for the first time (FPDEs). Without any transformation, the suggested approach can be used to solve fractional-order nonlinear Zakharov–Kuznetsov equations. The Caputo notion of the fractional-order derivative, whose values fall within the closed interval [0, 1], has been taken into consideration. The method's appeal is that it provides an approximate solution after just one iteration. The suggested method's numerical findings have been contrasted with those of the variational iteration method, residual power series method, and perturbation iteration method. Through tables and graphs, the proposed method's effectiveness and dependability are demonstrated.

Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations

In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variational homotopy perturbation iteration method (VHPIM) with a modified Riemann-Liouville derivative to solve the fuzzy fractional diffusion equation (FDE). Using this approach achieves a rapidly convergent sequence that approaches the exact solution of the equation. The proposed methods are investigated based on fuzzy fractional derivatives with some illustrative examples. The results reveal that the schemes are highly effective for obtaining the solutions to fuzzy fractional partial differential equations.

Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques

In this article, three different techniques, the Fractional Perturbation Iteration Method (FPIA), Fractional Successive Differentiation Method (FSDM), and Fractional Novel Analytical Method (FNAM), have been introduced. These three iterative methods are applied on different types of Electrical RLC-Circuit Equations of fractional-order. The fractional series approximation of the derived solutions can be established by using the obtained coefficients. These three algorithms handle the problems in a direct manner without any need for restrictive assumptions. The comparison displays an agreement between the obtained results. The beauty of this paper lies in the error analysis between the exact solution and approximate solutions obtained by these three methods which prove that the Approximate Solution obtained by FNAM converge very rapidly to the exact solution.

Numerical approximations for a class of nonlinear higher order singular boundary value problem by using homotopy perturbation and variational iteration method

In this work, we focus on the non-linear fourth order class of singular boundary value problem which contains the parameter λ . We convert this non-linear differential equation into third order non-linear differential equation. The third order problem is singular, non self adjoint, and nonlinear. Moreover, depending upon λ , it admits multiple solutions. Hence, it is too difficult to capture these solutions by any discrete method such as finite difference and so forth. Here we develop an iterative technique with the help of homotopy perturbation method and variational iteration method in a suitable way. Convergence of this series solution is studied in a novel way. We compute these solutions numerically. For small positive values of λ , singular BVP has two solutions while solutions can not be found for large positive values of λ . Furthermore, we also find dual solutions for λ < 0 .

Approximate solution of the electrostatic nanocantilever model via optimal perturbation iteration method

Approximate solution of the electrostatic nanocantilever model via optimal perturbation iteration method

On the key role of crack surface area on the lifetime of arbitrarily shaped flat cracks

Accurate fatigue lifetime prediction is a central element in many industries. Often, a damage tolerant approach is used to ensure that some given imperfections are safe. In this context, these lifetime predictions are frequently made using simplified crack geometries leading to costly conservatism. In particular for a closed crack front, a circular or elliptical crack enclosing the real defect is generally used leading to an underestimation of the real lifetime. Using an iterative perturbation method to simulate the fatigue growth of many tensile complexly shaped cracks, we show that in the absence of any additional complexity, (i) the front quickly becomes circular; (ii) its evolution with the number of loading cycles can be obtained analytically using a circular crack of same area as the original distorted defect. In practice, it means that replacing a complex crack front shape with this simple, analytically solvable configuration is an effective way to improve predictions and that it is not useful to invoke more complex shapes such as elliptical cracks. These take-home messages are illustrated by an example from the aviation industry.

Accelerated homotopy perturbation iteration method for a non-smooth nonlinear ill-posed problem

In this paper, a fast iterative algorithm based on J-order homotopy perturbation method is proposed for the nonlinear ill-posed problem whose forward operator is not Gâteaux differentiable. The Bouligand subderivative of the forward operator is utilized to replace the Fréchet derivative in iteration system. The sequential subspace optimization technique is introduced to accelerate the convergence speed, which regards the correction term of homotopy perturbation as multiple search directions to update the new iterate. To this end, the current iteration is sequentially projected to the stripes whose width is determined by search directions, the nonlinearity of the forward operator and noise level. We present the convergence analysis based on the asymptotic stability estimates and a generalized tangential cone condition. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey–predator model with stability analysis

Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey–predator model with stability analysis

Mesoscale computational prediction of lightweight, thermally conductive polymer nanocomposites containing graphene-wrapped hollow particle fillers

Heat removal has become an increasingly crucial issue for microelectronic chips due to increasingly high speed and high performance. One solution is to increase the thermal conductivity of the corresponding dielectrics. However, traditional approach to adding solid heat conductive nanoparticles to polymer dielectrics led to a significant weight increase. Here we propose a dielectric polymer filled with heat conductive hollow nanoparticles to mitigate the weight gain. Our mesoscale simulation of heat conduction through this dielectric polymer composite microstructure using the phase-field spectral iterative perturbation method demonstrates the simultaneous achievement of enhanced effective thermal conductivity and the low density. It is shown that additional heat conductivity enhancement can be achieved by wrapping the hollow nanoparticles with graphene layers. The underlying mesoscale mechanism of such a microstructure design and the quantitative effect of interfacial thermal resistance will be discussed. This work is expected to stimulate future efforts to develop light-weight thermal conductive polymer nanocomposites.

An extension of optimal auxiliary function method to fractional order high dimensional equations

In this article, the new approach called, the optimal auxiliary function method (OAFM) has been extended with the help of the fractional complex transform method (FCTM) to fractional order partial differential equations (PDEs). The beauty of the method is that, there is no need for small or large parameter assumption in the problem. Similarly, the proposed method gives an approximate solution after only one itearion. To test the proposed method, we take the time-fractional order Zakharov-Kuznetsov equations as a test example. The computed results of the suggested method are compared with the new iterative method (NIM), perturbation iteration method (PIA), and Residual power series method (RPSM). The accuracy and effectiveness of the proposed method can be proven by numerical and graphical results.

Phase-controlled optical Kerr effect in a microwave-driven X-type atomic system

We theoretically investigate the first- and third-order probe response of a five-level X-type atomic system coupled by the microwave field. The analytical solutions describing the phase dependence of linear/nonlinear probe coherence are obtained by invoking the iterative perturbation method. We find that for both stationary and moving atoms, EIT and Kerr nonlinear behavior of our considered system can be manipulated by altering the relative phase ( $$\phi $$ ) of the applied e.m. fields. The appropriate choice of $$\phi $$ gives rise to colossal Kerr nonlinearity accompanied by the reduced absorption. Further, we analyzed the effect of wavelength mismatch on the linear/nonlinear optical response for three different regimes of wavelength. Our results demonstrate that out of the three wavelength mismatch regimes, maximum Kerr index is obtained when probe field’s wavelength is chosen to be larger than that of the upper two control fields, i.e., $$\lambda _p=\lambda _{c_1}>\lambda _{c_2}\approx \lambda _{c_3}$$ . Besides providing the largest Kerr effect, this scheme is easy to implement experimentally.

A new efficient technique for solving modified Chua\u2019s circuit model with a new fractional operator

Chua’s circuit is an electronic circuit that exhibits nonlinear dynamics. In this paper, a new model for Chua’s circuit is obtained by transforming the classical model of Chua’s circuit into novel forms of various fractional derivatives. The new obtained system is then named fractional Chua’s circuit model. The modified system is then analyzed by the optimal perturbation iteration method. Illustrations are given to show the applicability of the algorithms, and effective graphics are sketched for comparison purposes of the newly introduced fractional operators.

A modified analytical model of the alkali-metal atomic magnetometer employing longitudinal carrier field* *Project supported by the Hunan Graduate Research and Innovation Project (Grant No. CX2018B009), the Natural Science Foundation of Hunan (Grant No. 2018JJ3608), the Research Project of National University of Defense Technology (Grant Nos. ZK170204 and ZZKY-YX-07-02), and the National Natural Science Foundation of China (Grant Nos. 61671458

Alkali-metal atomic magnetometers employing longitudinal carrier magnetic field have ultrahigh sensitivity to measure transverse magnetic fields and have been applied in a variety of precise-measurement science and technologies. In practice, the magnetometer response is not rigorously proportional to the measured transverse magnetic fields and the existing fundamental analytical model of this magnetometer is effective only when the amplitudes of the measured fields are very small. In this paper, we present a modified analytical model to characterize the practical performance of the magnetometer more definitely. We find out how the longitudinal magnetization of the alkali metal atoms vary with larger transverse fields. The linear-response capacity of the magnetometer is determined by these factors: the amplitude and frequency of the longitudinal carrier field, longitudinal and transverse spin relaxation time of the alkali spins and rotation frequency of the transverse fields. We give a detailed and rigorous theoretical derivation by using the perturbation-iteration method and simulation experiments are conducted to verify the validity and correctness of the proposed modified model. This model can be helpful for measuring larger fields more accurately and configuring a desirable magnetometer with proper linear range.

Guaranteed distributed machine learning: Privacy-preserving empirical risk minimization.

Distributed learning over data from sensor-based networks has been adopted to collaboratively train models on these sensitive data without privacy leakages. We present a distributed learning framework that involves the integration of secure multi-party computation and differential privacy. In our differential privacy method, we explore the potential of output perturbation and gradient perturbation and also progress with the cutting-edge methods of both techniques in the distributed learning domain. In our proposed multi-scheme output perturbation algorithm (MS-OP), data owners combine their local classifiers within a secure multi-party computation and later inject an appreciable amount of statistical noise into the model before they are revealed. In our Adaptive Iterative gradient perturbation (MS-GP) method, data providers collaboratively train a global model. During each iteration, the data owners aggregate their locally trained models within the secure multi-party domain. Since the conversion of differentially private algorithms are often naive, we improve on the method by a meticulous calibration of the privacy budget for each iteration. As the parameters of the model approach the optimal values, gradients are decreased and therefore require accurate measurement. We, therefore, add a fundamental line-search capability to enable our MS-GP algorithm to decide exactly when a more accurate measurement of the gradient is indispensable. Validation of our models on three (3) real-world datasets shows that our algorithm possesses a sustainable competitive advantage over the existing cutting-edge privacy-preserving requirements in the distributed setting.