Accuracy Of Approximate Solution Research Articles

Calculation of the spherical elements of non-uniform thickness for structures with holes based on the variational RVR-method

This paper proposes a theoretically substantiated and universal new method to calculate the three-dimensional stressed-strained state of the statically loaded multi-link orthotropic shell of arbitrary thickness, made of heterogeneous material (a composite). The numerical-analytical RVR method used in this work is based on the Reissner principle, Vekua method, the R-function theory, as well as the algorithm of two-way assessment of the accuracy of approximate solutions to variational problems. In contrast to the classical principles by Lagrange and Castigliano, the application of the mixed variational Reissner principle yields an increase in the accuracy of solving boundary-value problems due to the independent variation of the displacement vector and the stress tensor. Vekua method makes it possible, as a result of expanding the desired functions into a Fourier series based on Legendre polynomials, to replace a solution to the three-dimensional problem with a regular sequence of solutions to the two-dimensional problems in the process of refining the models of shells. The R-function theory that takes into consideration, at the analytical level, the geometric information on boundary-value problems for multi-relationship regions is necessary to build the structures of solutions that accurately meet different boundary conditions. When studying spatial boundary-value problems, the constructed algorithm for a two-way integrated assessment of the accuracy of approximate solutions makes it possible to automate the search for such a number of approximations at which the process of solutions’ convergence becomes persistent. For an orthotropic spherical shell made from the material of non-uniform thickness and weakened by the pole holes, the RVR-method capabilities are shown on the numerical examples of solving the relevant boundary-value problems. The results of the reported research have been discussed, as well as the features typical of the new method, which could be effectively applied when designing responsible shell-type elements of structures in the different sectors of modern industry

Boundary layer analysis for a 2-D Keller-Segel model

Abstract We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.

New operational ratios and their application to non-stationary tasks for rods based on S.P. Timoshenko theory

Relevance. In order to study transient wave processes of deformation in rods on the basis of S.P. Timoshenko theory, it is necessary to have accurate analytical solutions to non-stationary problems in general form. Each accurate solution within this analytical model is an accurate description of the real process, serves as a criterion in assessing the accuracy of approximate solutions. When using operational calculus to analyze traveling waves, it is the inverse Laplace - Carson transformation that poses the greatest difficulty. It follows from the published works that the available solutions to some private problems either have a structure that does not allow to judge the main features of the investigated process, or their efficiency in calculations is achieved only in some rather limited areas of coordinate and time. This problem, which requires resolution, determined the purpose of this article. The aim of the work. The article is devoted to the development of new operational ratios and their application to the construction of accurate analytical solutions to the non-stationary problems of S.P. Timoshenko's theory for rods in a general form, in a physically visible and convenient form for practical calculations. Methods. The work uses methods of function theory of complex variable, operational calculus based on the integral Laplace - Carson transformation, methods of structure dynamics. Results. In general form three types of non-stationary tasks for semi-infinite rod based on Timoshenko theory are formulated. New operational ratios have been obtained. Based on these ratios, a method of inverse transformation without using a general conversion formula has been developed. Solutions of problems are recorded in the form of integrals from Bessel functions and, unlike solutions available in the literature, clearly show the wave nature of the studied processes, have a visual and compact appearance. An example of calculation is reviewed.

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

Computational Framework Concerning the Formulation of Maximum Work Principle used in Plasticity, Materials Forming and Tribology as a Consequence of a Variational Optimization Problem Defined From the Constructal Law

This research work proposes a fundamental application of the Constructal Theory developed by Prof. Adrian BEJAN of Duke University (USA) to prove on a mathematical point of view the “Principle” of Maximum Work, used by the theory of plasticity, material forming and tribology, as a consequence of the global solution defining a constrained variational optimization problem. According to the first and second law of thermodynamics, the principal law of Constructal Theory try to complete them with a quantitative prediction of the natural tendency of any finite size system to evolve towards an optimal space-time configuration minimizing the losses and the entropy generation simultaneously with a required maximum of the global entropy. In this sense, regarding a material plastic deformation characterising the forming processes, among all possible and admissible flow undergoing well-specified boundary conditions and loadings, the real one is the one who minimizes the sum of the dissipated power of volume deformation and surfaces friction. Thus, all the mechanical variables defining the real mechanical state (velocities, stresses, strain and strain rate) are those ones which minimize the total dissipated power. A variational minimization problem under a lot of defined constraints is then obtained. Using the Principle of Virtual Powers it can be shown finally that the “Principle” of Maximum Work, used particularly in metals plasticity, is obtained as a consequence of a minimization problem under constraints based on the Constructal Theory. This generalizes its application to any type of continuous media (metals, polymers, fluid, mushy state) and allows proving an equivalent form for the friction stresses occurring on contact interfaces. The convexity properties of both the plastic and the friction potential together with their normal rule properties can be also proven using the proposed mathematical framework. It is concluded that only the rheological and tribological flow laws associated with a potential become to satisfy the second thermodynamics principle completed with the Constructal Law. Analytical computations concerning plane and cylindrical crushing show the feasibility of the proposed minimization problem formulation to define material flow giving accurate approximate solutions. In order to valid the whole presented theory, comparisons are made using classical analyses based on the upper and lower bound theorems (obtained as consequences of the proposed optimization variational problem), the well-known slices method and a finite element modelling (FEM). A second application concerning the anisotropic formulation of a Coulomb friction law from a quadratic convex tribologic potential will be presented to define contact evolution during rectilinear sliding of a circular pion on a plane laminated thick plate surface along different orientations.

Solutions of the Two-Wave Interactions in Quadratic Nonlinear Media

In this paper, we propose a reliable treatment for studying the two-wave (symbiotic) solitons of interactions in nonlinear quadratic media. We investigate the Schauder’s fixed point theorem for proving the existence theorem. Additionally, the uniqueness solution for this system is proved. Also, a highly accurate approximate solution is presented via an iteration algorithm.

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance.

An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing n,m for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy.

Recovery of timewise‐dependent heat source for hyperbolic PDE from an integral condition

The inverse problem of recovering the timewise‐dependent heat source coefficient along with the temperature in a second‐order hyperbolic equation with mixed derivative and with initial and Neumann boundary conditions and integral measurement is, for the first time, numerically investigated. The inverse problem considered in this paper has a unique solution. However, it is an ill‐posed problem by being sensitive to noise. The one‐dimensional inverse problem is discretized using the FDM and recast as a nonlinear least‐squares minimization of Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. The present numerical results demonstrate that accurate and stable approximate solutions have been attained.

A new multistage Parker-Sochacki method for solving the Troesch’s problem

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of in the range [20;500].

Spectrally accurate approximate solutions and convergence analysis of fractional Burgers\u2019 equation

In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

Asymptotic Solution for a Class of Epidemic Contagion Ecological Nonlinear Systems

In this paper, a class of systems for epidemic contagion is considered. An epidemic virus ecological model is described. Using the generalized variation iteration method, the corresponding approximate solution to the nonlinear system is obtained and the method for this approximate solution is pointed out. The accuracy of approximate solution is discussed, and it can control the epidemic virus transmission by using the parameters of the system. Thus, it has the value for practical application.

Error analysis and numerical solution of Burgers–Huxley equation using 3-scale Haar wavelets

In this article, we develop an efficient and accurate numerical scheme based on the Crank–Nicolson finite difference method and Haar wavelet analysis to evaluate the numerical solution of the Burgers–Huxley equation. The present method is extended form of Haar wavelet 2D scaling which shows that it is reliable for solving nonlinear partial differential equations. The numerical results are more accurate than other existing methods available in the literature and very close to the exact solution. The Haar basis function is generated from multi-resolution analysis and used to evaluate fast and accurate approximate solutions on the collocation points. The convergence of the proposed method is demonstrated by its error analysis. We compared numerical solutions with the exact solutions and solutions available in the literature. The proposed method is found to be straight forward, accurate with small computational cost and can be easily implemented in mathematical software MATLAB.

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.

Reproducing Kernel Method for Solving Nonlinear Oscillators Under Damping Effect

The current paper is devoted to introducing a semi-analytical technique for the numerical solution of nonlinear oscillators under the damping effect by using the reproducing kernel Hilbert space method. The proposed scheme is a hybrid of the reproducing kernel Hilbert space and the perturbation methods. This method finds solutions to periodic and nonlinear problems by converting to a system of linear differential equations. Finally, some test problems are given to illustrate the validity of the novel algorithm. Also, the presented numerical results using the method are compared with the numerical solutions of the fourth-order Runge–Kutta method. The numerical results reported show that the present method can provide very accurate approximate solutions.

Numerical study of multidimensional fractional time and space coupled Burgers’ equations

This paper declares a new spectral collocation technique to provide accurate approximate solutions of the one- and two-dimensional time and space fractional coupled Burgers’ equations (TSFCBEs) in which the fractional derivatives are defined according to Caputo’s definition. The suggested method is based on the shifted Gegenbauer polynomials (SGPs) for approximating the solution of TSFCBEs. The suggested technique reduces the considered problems to the solution of nonlinear algebraic equations (NLAEs). Moreover, the accuracy and reliability of the proposed method are confirmed through numerical examples. Finally, the obtained numerical results are compared with those previously reported in the literature.

A novel algorithm based on the Legendre wavelets spectral technique for solving the Lane–Emden equations

In this research, we present an iterative spectral method for the approximate solution of a class of Lane–Emden equations. In this procedure, we initially extend the Legendre wavelet which is appropriate for any time interval. Thereafter, the Guass-Legendre collection points of the Legendre wavelet are acquired. Employing this new approach, the iterative spectral technique converts the differential equation to a set of algebraic equations which diminishes the computational costs effectively. By solving the obtained algebraic equations, an accurate approximate solution for the assumed Lane–Emden equation is achieved. The present technique is validated by solving a number of Lane–Emden problems and are compared with other existing methods. The numerical simulations demonstrate that the new algorithm is simple and it has highly accuracy.

Stability Analysis of a Parametric Duffing Oscillator

The Duffing oscillator represents an important example to describe, mathematically, the nonlinear behavior of several phenomena in physics and engineering. In view of its diverse applications in science and engineering, the current work investigates the stability analysis of the parametric Duffing oscillator. This equation represents a second-order ordinary differential equation with cubic nonlinearity and periodic coefficient. A coupling of the homotopy perturbation method (HPM) and Laplace transform (LT), an approximate solution is derived. The HPM is adapted to find another accurate approximate solution. The latter analysis reveals the exact solution of the cubic Duffing equation. In addition, an expanded frequency parameter is achieved to find another approximate periodic solution. Therefore, this method is exercised to govern the stability criteria of the problem. Finally, the multiple time scales with the HPM is used to judge the stability criteria. The analyses include the resonance as well as nonresonance cases. Numerical estimations are performed to confirm, graphically, the perturbed solutions together with the stability examination. It is shown that the damped parameter and the cubic stiffness parameter have a destabilizing influence. In contrast, the natural and parametric frequencies are of stabilizing influences.

Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays

As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.