Accurate Analytical Approximate Solutions Research Articles

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>

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

Highly Accurate Analytical Approximate Solutions of the Electrostatically Actuated Parallel Plates

Highly Accurate Analytical Approximate Solutions of the Electrostatically Actuated Parallel Plates

Nonlinear analysis of Euler beams resting on a tensionless soil with arbitrary configurations

BackgroundThe nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. The exact analytical solutions of the nonlinear problem are rather complicated. The main difficulty is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution, although being needed to get the solution itself. In the current work, semi-analytical solutions are derived using the Rayleigh–Ritz method. The principle of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Accurate high-order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied automatically. The methodology is designed to accommodate arbitrary configurations for the load distribution and the beam properties.ResultsExact solutions are revised briefly to verify the semi-analytical solutions in terms of deflection, bending moment, and shear. Semi-analytical solutions for constant beam properties including various support conditions and load distributions are verified. Convergence of high-order semi-analytical solutions is illustrated for cases including one and two contact points. A parametric study is provided to illustrate the effect of soil stiffness on the contact length. The case of a finite beam with free ends is considered. The semi-analytical solutions for variable beam moment of inertia are provided and verified.ConclusionsHighly accurate semi-analytical solutions can be obtained for the problem considered using the Rayleigh–Ritz method along with a symbolic manipulator. Arbitrary load and support configurations can be modeled, and the locations of lift-off points are well predicted. The semi-analytical solutions are extremely valuable for cases of variable moment inertia since exact solutions are rather rare.

An Analytical Approximation for the Ionomer Film Model in PEMFC

The ionomer film and its transport resistances for oxygen are considered to be an important aspect for PEMFC performance. Ionomer film sub-models are therefore frequently used in PEMFC modeling to account for this effect. Mathematically these are expressed by a non-linear equation for the oxygen concentration, which depending on the reaction order cannot be solved analytically. Typically, a numerical solution of this equation, e.g., using the Newton-method is needed. Here, we derive a highly accurate approximate analytical solution for the ionomer film model. This enables faster computation, which is particularly important for computationally demanding higher dimensional PEMFC models.

Toward an efficient approximate analytical solution for 4-compartment COVID-19 fractional mathematical model

With the recent trend in the spread of coronavirus disease 2019 (Covid-19), there is a need for an accurate approximate analytical solution from which several intrinsic features of COVID-19 dynamics can be extracted. This study proposes a time-fractional model for the SEIR COVID-19 mathematical model to predict the trend of COVID-19 epidemic in China. The efficient approximate analytical solution of multistage optimal homotopy asymptotic method (MOHAM) is used to solve the model for a closed-form series solution and mathematical representation of COVID-19 model which is indeed a field where MOHAM has not been applied. The equilibrium points and basic reproduction number (R0) are obtained and the local stability analysis is carried out on the model. The behaviour of the pandemic is studied based on the data obtained from the World Health Organization. We show on tables and graphs the performance, behaviour, and mathematical representation of the various fractional-order of the model. The study aimed to expand the application areas of fractional-order analysis. The results indicate that the infected class decreases gradually until 14 October 2021, and it will still decrease slightly if people are being vaccinated. Lastly, we carried out the implementation using Maple software 2021a.

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

In providing an accurate approximate analytical solution to the non-linear system of fractional-order susceptible-infected-recovered epidemic model (FOSIREM) of childhood disease has been a challenge, because no norm to guarantees the convergence of the infinite series solution. We compute an accurate approximate analytical solution using the optimal homotopy asymptotic method (OHAM). The fractional differential equations operator (FDEO) is given as conformable derivative operator (CDO). We show the basic idea of the proposed method, the CDO sense, equilibrium points, local asymptotic stability, reproduction number, and the convergence analysis of the proposed method. Numerical results and comparisons with other approximate analytical methods are given to validate the efficiency of the method. The proposed method speedily converges to the exact solution as the fractional-order derivative approaches 1, proved as an excellent tool for solving, and predicting the model.

Approximate Analytical Solutions for Systems of Fractional Nonlinear Integro-Differential Equations Using the Polynomial Least Squares Method

We employ the Polynomial Least Squares Method as a relatively new and very straightforward and efficient method to find accurate approximate analytical solutions for a class of systems of fractional nonlinear integro-differential equations. A comparison with previous results by means of an extensive list of test-problems illustrate the simplicity and the accuracy of the method.

ANALYTICAL SOLUTION FOR NONLINEAR INTERACTION OF EULER BEAM RESTING ON A TENSIONLESS SOIL

The nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. Exact analytical solutions of the challenging problem are rather complicated. The basic obstacle is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution although being needed to get the solution itself. In the current work, solutions are derived using the approximate Rayleigh-Ritz method. The principal of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied implicitly. Results are presented for various cases, including clamped support and free end condition. Various distributed loading conditions are analyzed. Exact solutions are revised briefly. Accurate high order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. The convergence of approximate solutions towards the exact solutions is verified. For each case detailed results of deflection, bending moment and shear are presented.

The nonlinear viscoelastic response of suspensions of rigid inclusions in rubber: I—Gaussian rubber with constant viscosity

A numerical and analytical study is made of the macroscopic or homogenized viscoelastic response of suspensions of rigid inclusions in rubber under finite quasistatic deformations. The focus is on the prototypical case of random isotropic suspensions of equiaxed inclusions firmly embedded in an isotropic incompressible Gaussian rubber with constant viscosity. From a numerical point of view, a robust scheme is introduced to solve the governing initial–boundary-value problem based on a conforming Crouzeix–Raviart finite-element discretization of space and a high-order accurate explicit Runge–Kutta discretization of time, which are particularly well suited to deal with the challenges posed by finite deformations and the incompressibility constraint of the rubber. The scheme is deployed to generate sample solutions for the basic case of suspensions of spherical inclusions of the same (monodisperse) size under a variety of loading conditions. From a complementary point of view, analytical solutions are worked out in the limits: (i) of small deformations, (ii) of finite deformations that are applied either infinitesimally slowly or infinitely fast, and (iii) when the rubber loses its ability to store elastic energy and reduces to a Newtonian fluid. Strikingly, in spite of the fact that the underlying rubber matrix has constant viscosity, the solutions reveal that the viscoelastic response of the suspensions exhibits an effective nonlinear viscosity of shear-thinning type. The solutions further indicate that the viscoelastic response of the suspensions features the same type of short-range-memory behavior — as opposed to the generally expected long-range-memory behavior — as that of the underlying rubber. Guided by the asymptotic analytical results and the numerical solutions, a simple yet accurate approximate analytical solution for the macroscopic viscoelastic response of the suspensions is proposed.

Highly Accurate Analytical Approximate Solutions to Mixed-Parity Duffing Equation

In this paper, a highly accurate analytical approximation solution method to a class of mixed-parity Duffing equation is proposed. The system may be attributed to the free vibration of laminated anisotropic plates or the simplified problem of particle vibration. The system is first analyzed qualitatively and subsequently, the analytic approximate periodic solution within the parametric range is constructed. The first approximate solution with a suitable initial condition can be obtained by using the proposed method. Subsequently, higher precision approximate solutions are constructed by combining Newton’s method and harmonic balance method. These analytical solutions have excellent approximation accuracy as verified by numerical solutions derived from exact analytical expressions.

The Novel Integral Homotopy Expansive Method

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

Accurate analytical approximation to post-buckling of column with Ramberg−Osgood constitutive law

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

A new mathematical technique for analysis of internal viscoplastic flows through rectangular ducts

The flow of Bingham materials inside rectangular channels is characterized by a plug region located in the central region of the conduit and four dead zones in the vicinity of the corners of the geometry (i.e., the unyielded regions) that grow as the Bingham number is increased. In this paper, a semianalytical technique is presented for solving the flow of Bingham yield-stress materials inside straight rectangular ducts. To capture the topological shapes of the yielded and unyielded regions, a “squircle” mapping approach is proposed and the interfaces between the yielded and unyielded regions are captured based on minimizing the variational formulation for the velocity principle. The nonlinear governing equations are consequently solved using the Ritz method. To find the optimized solution, a genetic algorithm method is implemented, applying bounded constraints and a sufficient number of populations. The presented results are benchmarked against numerical works available in literature, revealing that the presented solution can accurately capture the positions of both the plug and dead regions. The critical Bingham number at which the plug region meets the dead regions is reported for different aspect ratios. The velocity profiles and the friction factor of flow with different Bingham numbers and aspect ratios are investigated in detail. Following this, in the limiting case in which the Bingham number is sufficiently small, an alternative, simpler analytical approach using elliptical mapping is also proposed. The authors believe that the proposed method could be employed as a useful tool to obtain fast, accurate approximate analytical solutions for similar classes of problem.

Fluctuation and Frequency of the Oscillators with Exponential Spring Using Accurate Approximate Analytical Solutions

This paper examines the fluctuation and frequency of the motion of a homogeneous solid on a horizontal plane on an exponential crisscross spring. The governing equations of this oscillator are investigated using a modified Homotopy Perturbation method, Energy balance method and artificial parameter–Linstedt–Poincare’ method. A comparison of solutions shows that the fluctuation and frequency of the system solved by the presented methods are effective with a minor error of 0.1%.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

More Accurate Approximate Analytical Solution of Nonlinear Oscillators

A more accurate approximate analytical solution of nonlinear oscillator has been provided by using modified energy balance method. The results have been compared with those obtained by homotopy perturbation method (HPM) and a new homotopy perturbation method (NHPM). It has been observed that the results obtained by proposed method are better than those of HPM as well as its new version. Moreover the method provides better results than other existing methods. Cite as M. H. U. Molla, M. Z. Alam, Nazmul Sharif, & M. S. Alam. (2020). More Accurate Approximate Analytical Solution of Nonlinear Oscillators. Journal of Applied Mathematics and Statistical Analysis, 1(3), 1–7. http://doi.org/10.5281/zenodo.4311134

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

A Generalization of Lindstedt–Poincaré Perturbation Method to Strongly Mixed-Parity Nonlinear Oscillators

In this paper, generalization for the Lindstedt–Poincare perturbation method for nonlinear oscillators to a class of strongly mixed-parity oscillating system is established. In this extended and enhanced approach, two new odd nonlinear oscillators are introduced in terms of the mixed-parity oscillator. By combining the analytical approximate solutions corresponding to the two new systems, the accurate approximate solutions of the original mixed-parity oscillator are obtained. Comparing with the existing methods such as the perturbation method, the new solution methodology for the singular nonlinear system introduced is not only simple, but the combinatory solution is straight forward and it yields very accurate and physically insightful solutions. Using two typical examples, we demonstrated that this new proposed approach is capable of establishing highly accurate approximate analytical frequency and periodic solutions for small as well as large amplitude of oscillation. The new analytical methodology established will potentially shed new insights to the physical interpretations of strongly nonlinear oscillators including optoelectronic oscillators, pendulums and spring-masses.

Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs.