Optimal Convergence-control Parameters Research Articles

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

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.

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.

Dynamics of SEIR epidemic model by optimal auxiliary functions method.

The aim of the present work is to establish an approximate analytical solution for the nonlinear Susceptible, Exposed, Infected, Recovered (SEIR) model applied to novel coronavirus COVID-19. The mathematical model depending of five nonlinear differential equations, is studied and approximate solutions are obtained using Optimal Auxiliary Functions Method (OAFM). Our technique ensures a fast convergence of the solutions after only one iteration. The nonstandard part of OAFM is described by the presence of so-called auxiliary functions and of the optimal convergence-control parameters. We have a great freedom to select the auxiliary functions and the number of optimal convergence-control parameters which are optimally determined. Our approach is independent of the presence of small or large parameters in the governing equations or in the initial/boundary conditions, is effective, simple and very efficient.

A new analytical approach for the research of thin‐film flow of magneto hydrodynamic fluid in the presence of thermal conductivity and variable viscosity

AbstractIn this endeavor thin‐film flow of magneto hydrodynamic (MHD) fluid in the presence of thermal conductivity and variable viscosity over a porous steady stretching surface with a magnetic field and radioactive heat fluctuation is studied. In this phenomenon the viscosity vary inversely and thermal conductivity directly with temperature. The nonlinear coupled differential equations for the velocity and temperature profiles are achieved and solved by using a new analytical approach, called 3rd form of Optimal Homotopy Asymptotic Method (OHAM‐3). The proposed technique consists of initial guess, embedding parameter, optimal convergence control parameters, auxiliary functions and homotopy. Numerical (ND‐Solve Method) solutions are also achieved and compared with the results gained by the proposed method. The fast convergence of the applied new method and the influence of different physical nondimensional parameters on the velocity and temperature profiles are mainly focused in this research.

Fractional transform methods for coupled system of time fractional derivatives of non-homogeneous Burgers’ equations arise in diffusive effects

The coupled system of time fractional derivatives of non-homogeneous Burgers’ equations is solved both analytically and numerically. The approximate analytical solutions of power series type are obtained by the fractional homotopy analysis transform method, which is verified numerically by the coupled fractional reduced differential transform method. Particularly, the present analytical solutions are compared with the results available in the literature for several special cases, and very excellent agreement was found. An error analysis is done to compute the average squared residual errors of those analytical approximations for other cases. The present analysis shows that the proposed technique has very excellent efficiency to give solutions with high precision. On the other hand, it is found that the optimal convergence control parameter derived here not only controls the convergence of the present solutions but also leads to identifying multiple solutions.

Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic

The goal of this paper is to present a new scheme based on the stochastic arithmetic (SA) to find the optimal convergence control parameter, the optimal iteration and the optimal approximation of the homotopy analysis method (HAM). This scheme is called the CESTAC1 method. Also, the CADNA2 library is applied to implement the CESTAC method on the proposed algorithms. CADNA is able to present a new, robust and valid environment to implement the HAM and optimize the results. By using this method, not only the optimal auxiliary control parameter can be computed but also the unnecessary iterations can be neglected and optimal step of this method is achieved. The main theorems are presented to guarantee the validity and accuracy of the HAM. Different kinds of integral equations such as singular and first kind are considered to find the optimal results by applying the proposed algorithms. The numerical results show the importance and efficiency of the SA in comparison with the floating-point arithmetic (FPA).

Approximate solution of space and time fractional higher order phase field equation

This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

The Optimal Homotopy Analysis Method for solving linear optimal control problems

In this paper, an Optimal Homotopy Analysis Method (Optimal HAM) is applied to solve the linear optimal control problems (OCPs), which have a quadratic performance index. This approach contains at most two convergence-control parameters which depend on the control system and is computationally rather efficient. A squared residual error for the system is defined, which can be used to find the unknown optimal convergence-control parameters by using Mathematica package BVPh (version 2.0). The results of comparisons among the proposed method, the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), the differential transform method (DTM) and the homotopy analysis method (HAM) provide verification for the validity of the proposed approach. Moreover, numerical results are presented by several examples involving scalar and 2nd-order systems to clarify the efficiency and high accuracy of the proposed approach.

Optimal q-homotopy analysis method for time-space fractional gas dynamics equation

It is well known that the homotopy analysis method is one of the most efficient methods for obtaining analytical or approximate semi-analytical solutions of both linear and non-linear partial differential equations. A more general form of HAM is introduced in this paper, which is called Optimal q-Homotopy Analysis Method (Oq-HAM). It has better convergence properties as compared with the usual HAM, due to the presence of fraction factor associated with the solution. The convergence of q-HAM is studied in details elsewhere (M.A. El-Tawil, Int. J. Contemp. Math. Sci. 8, 481 (2013)). Oq-HAM is applied to the non-linear homogeneous and non-homogeneous time and space fractional gas dynamics equations with initial condition. An optimal convergence region is determined through the residual error. By minimizing the square residual error, the optimal convergence control parameters can be obtained. The accuracy and efficiency of the proposed method are verified by comparison with the exact solution of the fractional gas dynamics equation. Also, it is shown that the Oq-HAM for the fractional gas dynamics equation is equivalent to the exact solution. We obtain graphical representations of the solutions using MATHEMATICA.

Improved First-Order Approximate Models of Temperature and Thermal Stresses for Online Fatigue Monitoring

Online monitoring of temperature and thermal stresses is an important way to ensure the safety of power plants considering fatigue and creep damages. The effect of online monitoring is determined by the accuracy and calculating time of monitoring models. In this paper, the improved first-order analytical models of temperature and thermal stresses considering temperature-dependent material properties have been derived by using homotopy analysis method (HAM) and superposition principle. The optimal convergence control parameters are obtained by calculating the mean-square residual errors. The validity and accuracy of the proposed models were proved by results comparisons with finite element method (FEM) and artificial parameter method.

AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

This paper describes a new problem-solving mentality of finding optimal parameters in optimal homotopy analysis method (optimal HAM). We use particle swarm optimization (PSO) to minimize the exact square residu- al error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has fi- nite convergence-control parameters. Two nonlinear fractional-order differen- tial equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Mean- while, we give a sufficient condition for convergence of the optimal HAM for solving fractional-order equation, and try to put forward a new calculation method for the residual error.

Casson fluid flow with variable thermo-physical property along exponentially stretching sheet with suction and exponentially decaying internal heat generation using the homotopy analysis method

This article studies the motion of temperature dependent plastic dynamic viscosity and thermal conductivity of steady incompressible laminar free convective magnetohydrodynamic (MHD) Casson fluid flow over an exponentially stretching surface with suction and exponentially decaying internal heat generation. It is assumed that the natural convection is driven by buoyancy and space dependent heat generation. The viscosity and thermal conductivity of Casson fluid is assumed to vary as a linear function of temperature. By using suitable transformation, the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear coupled ordinary differential equations and solved by the Homotopy analysis method. A new kind of averaged residual error is adopted and used to find the optimal convergence control parameter. A parametric study is performed to illustrate the influence of Prandtl number, Casson parameter, temperature dependent viscosity, temperature dependent thermal conductivity, Magnetic parameter and heat source parameter on the fluid velocity and temperature profiles within the boundary layer. The flow controlling parameters are found to have a profound effect on the resulting flow profiles.

A spectral method for the electrohydrodynamic flow in a circular cylindrical conduit

This paper presents a combination of the hybrid spectral collocation technique and the spectral homotopy analysis method (SHAM for short) for solving the nonlinear boundary value problem (BVP for short) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. The accuracy of the present solution is found to be in excellent agreement with the previously published solution. The authors use an averaged residual error to find the optimal convergence-control parameters. Comparisons are made between SHAM generated results, results from literature and Matlab ode45 generated results, and good agreement is observed.

Singularly perturbed homotopy analysis method applied to the pressure driven flame in porous media

In this paper we combined the homotopy analysis method (HAM) and the method of integral manifold (MIM) for the problem of a pressure-driven flame in an inert medium filled with a flammable gaseous mixture. In order to apply the MIM, we have rewritten the non-dimensional model in the form of a singular perturbed system (SPS) of ordinary differential equations. Two different stages of the trajectory correspond to the sub-zones of the flame: (1) fast motion from the initial condition to the slow curve, and its interpretation as a preheat sub-zone and (2) the path that corresponds to a reaction sub-zone.By applying the HAM and the MIM we derived an analytical expression for the condition of ignition and also an analytical expression for the delay time. The assumption that the heat flux is continuous at the point where the fast and slow parts of the trajectory are glued enabled us to obtain an expression for the flame velocity. Our theoretical results are justified by the numerical simulations.An optimal value of the convergence control parameter is given through the square residual error. By minimizing the square residual error, the optimal convergence-control parameters can be obtained. The more quickly the residual error decreases to zero, the faster the corresponding homotopy series solution converges.

Heat transfer analysis of Williamson fluid over exponentially stretching surface

This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature (PEST) case and the prescribed exponential order heat flux (PEHF) case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method (OHAM). The optimal convergence control parameters are obtained, and the physical features of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.

Analytic approximation to nonlinear hydroelastic waves traveling in a thin elastic plate floating on a fluid

An analytic approximation method known as the homotopy analysis method (HAM) is applied to study the nonlinear hydroelastic progressive waves traveling in an infinite elastic plate such as an ice sheet or a very large floating structure (VLFS) on the surface of deep water. A convergent analytical series solution for the plate deflection is derived by choosing the optimal convergencecontrol parameter. Based on the analytical solution the effects of different parameters are considered. We find that the plate deflection becomes lower with an increasing Young’s modulus of the plate. The displacement tends to be flattened at the crest and be sharpened at the trough as the thickness of the plate increases, and the larger density of the plate also causes analogous results. Furthermore, it is shown that the hydroelastic response of the plate is greatly affected by the high-amplitude incident wave. The results obtained can help enrich our understanding of the nonlinear hydroelastic response of an ice sheet or a VLFS on the water surface.

Application of Optimal HAM for Finding Feedback Control of Optimal Control Problems

An optimal homotopy-analysis approach is described for Hamilton-Jacobi-Bellman equation (HJB) arising in nonlinear optimal control problems. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A kind of averaged residual error is defined. By minimizing the averaged residual error, the optimal convergence-control parameters can be obtained. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity. The closed-loop optimal control is obtained using the Bellman dynamic programming. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.

Solving a model for the evolution of smoking habit in Spain with homotopy analysis method

We obtain an approximated analytical solution for a dynamic model for the prevalence of the smoking habit in a constant population but with equal and different from zero birth and death rates. This model has been successfully used to explain the evolution of the smoking habit in Spain. By means of the Homotopy Analysis Method, we obtain an analytic expression in powers of time t which reproduces the correct solution for a certain range of time. To enlarge the domain of convergence we have applied the so-called optimal convergence-control parameter technique and the homotopy-Padé technique. We present and discuss graphical results for our solutions.

The improved homotopy analysis method for the Thomas–Fermi equation

The homotopy analysis method (HAM) is sharpened to solve the Thomas–Fermi equation. Some techniques are employed, including the use of asymptotic analysis to introduce proper transformation, and the use of optimal initial guess and optimal auxiliary linear operator to accelerate the convergence of homotopy approximations. The optimal convergence-control parameters are determined by the minimum of the squared residual error. As a result, the initial slop is provided with more-than-10-digit accuracy, which is far more accurate than the results obtained by other authors using the same method. It demonstrates the flexibility and power of the HAM equipped with these techniques.