Approximate Numerical Solutions Research Articles

On the formulation of a predictor–corrector method to model IVPs with variable‐order Liouville–Caputo‐type derivatives

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described.

Geometric Characteristics of Simply Connected Planar Sections With Application to Airfoils

The presented work is to develop a numerical computation program to determine the geometrical characteristics of arbitrary simply connected planar sections and applications to airfoils. In the literature, there are exact analytical solutions only for some simple geometry such as circular, rectangular and elliptical sections. Hence our interest is focused on the search of approximate numerical solutions for more complex sections used in aeronautics. The used method is to subdivide the section in the infinitesimal triangular section with a single observer within the area having a suitable position. The characteristics of any triangle, given by the positions of these three nodes are known in the literature. Using the principle of compound surfaces, one can determine the geometric characteristics of the airfoil surface. The analytic function of the airfoil boundary is obtained by using the cubic spline interpolation because the airfoil is given in the form of tabulated points. An error estimation is done to determine the accuracy of the numerical computation.

A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities

Abstract The primary goal of this research is to investigate the approximate numerical solution of variational inequalities using quasimonotone operators in infinite-dimensional real Hilbert spaces. In this study, the sequence obtained by the proposed iterative technique for solving quasimonotone variational inequalities converges strongly toward a solution due to the viscosity-type iterative scheme. Furthermore, a new technique is proposed that uses an inertial mechanism to obtain strong convergence iteratively without the requirement for a hybrid version. The fundamental benefit of the suggested iterative strategy is that it substitutes a monotone and non-monotone step size rule based on mapping (operator) information for its Lipschitz constant or another line search method. This article also provides a numerical example to demonstrate how each method works.

Fractional derivative modeling for characterizing stress relaxation properties of Hami melon during drying.

The viscoelastic properties of food materials will change significantly while drying is in progress, which greatly influences the food deformation caused by drying. This study aims to predict the viscoelastic mechanical behavior of Hami melon during drying using the fractional derivative model. To characterize the relaxation characteristics, based on the finite difference method, an improved Grünwald-Letnikov fractional stress relaxation model is proposed to derive an approximate discrete numerical solution of the relaxation modulus by applying the time fractional calculus. The Laplace transform method is used to verify the obtained results, and the equivalence of the two methods is proved. In addition, the stress relaxation tests prove that the fractional derivative model has a better prediction effect on the stress relaxation behavior of viscoelastic food than classical Zener model. The significant correlations between the fractional order and the stiffness coefficient and the moisture content is also studied. Which is negative correlation and positive correlation respectively.

Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi

Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.

New Numerical Results on Existence of Volterra–Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type

Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach.

Numerical Simulations for the Gross-Pitaevskii Equation

In this study the split-step Fourier method for the numerical simulation of the Gross-Pitaevskii equation. Approximate numerical solutions of the GrossPitaevskii equation are obtained by using Matlab software. It is shown that the proposed method improves the computational effort significantly. This improvement becomes more significant especially for large time evolutions.

Existence of Monotone Positive Solutions for Caputo–Hadamard Nonlinear Fractional Differential Equation with Infinite-Point Boundary Value Conditions

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.

An almost second-order robust computational technique for singularly perturbed parabolic turning point problem with an interior layer

The subject of this paper is the numerical investigation of a singularly perturbed parabolic problem with a simple interior turning point on a rectangular domain in the x−t plane with Dirichlet boundary conditions. At the location of the turning point, the solution to this class of problems has an internal layer. By decomposing the solution into a regular and singular component, we may first establish stronger bounds for the regular and singular components, as well as their derivatives. The fitted mesh finite difference scheme, which consists of a hybrid finite difference operator on a non-uniform Shishkin mesh in the space direction and a backward implicit Euler scheme on a uniform mesh in the time variable, is used to identify the approximate numerical solution. The accuracy of the resulting scheme is then improved in the temporal direction using the Richardson extrapolation scheme, which reveals that the extrapolated scheme is almost second-order uniformly convergent in both space and time variables when ɛ≤CN−1. Two numerical examples are used to demonstrate the efficacy and accuracy of the finite difference approaches mentioned.

An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem

The q-homotopy analysis method (q-HAM) in combine with the novel integral transform known as Elzaki transform (ET) leads to an efficient analytical technique called, the q-homotopy analysis Elzaki transform method (q-HAETM). In the present study, the two- dimensional advection–dispersion (AD) problem is investigated using an analytical technique q-HAETM. These equations are mainly used to describe the fate of pollutants in aquifers. The analytical solutions to the AD equations are more interesting since they serve as benchmarks against which numerical solutions can be compared. The novelty of the work is to discuss the two-dimensional (2D) solute transport problem in the fractional sense. The reliability and the efficiency of the considered algorithm are demonstrated by employing the 2D fractional solute transport problem. The solute concentration profile is shown in terms of surface plots. The comparison of the exact solution and the approximate solution is done by the 2D plots. The numerical approximate error solutions are presented for different fractional orders. q-HAETM offers us to modulate the range of convergence of the series solution using ℏ, called auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations in comparison with other existing techniques, the effectiveness and reliability of the considered technique are validated. The obtained findings show that the proposed method is very gratifying and examines the complex challenges that arise in science and innovation.

Continuum Model on (HYTXW) of Fungal Growth

In general the growth of fungi needs to be resolved until its goal becomes a correction and thus, we minimize cost, effort, time and money. There-fore, we arrived at a mathematical solution using some techniques. In this paper, we studied its growth behavior and the effect of each branch on the fungus, then we combined a number of branches represented as mathematical model as partial differential equations (PDEs), approximate numerical solutions, and some math-ematical steps, such as non-dimensionalisation, finding stability or steady state and representing it on phase plane, we found approximate results for these types using MATLAB’s codes such as Pplane8 and Pdepe [1,7].

Iterative Positive Solutions to a Coupled Riemann-Liouville Fractional q -Difference System with the Caputo Fractional q -Derivative Boundary Conditions

This paper is devoted to the existence of positive solutions for a nonlinear coupled Riemann-Liouville fractional q -difference system, with multistrip and multipoint mixed boundary conditions under Caputo fractional q -derivative. We obtain the existence of positive solutions and initial iterative solutions by the monotone iteration technique. Then, we also calculate the error limits of the numerical approximation solution by induction. In the end, two examples are given to illustrate the above research results, and in the second example, some graphs of the iterative solutions are also drawn to give a more intuitive sense of the iterative process.

Theoretical Simulation of the High–Order Harmonic Generated from Neon Atom Irradiated by the Intense Laser Pulse

Based on the strong field approximation theory and numerical solution of Maxwell’s propagation equations, the high–order harmonic is generated from a neon (Ne) atom irradiated by a high–intensity laser pulse whose central wavelength is 800 nm. In the harmonic spectrum, it is found that in addition to the odd harmonics of the driving laser, a new frequency peak appeared. By examining the time–dependent behavior of the driving laser, it is found that the symmetry of the laser field is broken. We demonstrated that these new spectrum peaks are caused by the intensity reduction and frequency blue shift of the high–intensity laser during propagation. Our results reveal that it is feasible to modulate the harmonics of the specific energy to produce high–intensity harmonic emission by changing the gas density and the position of the gas medium interacting with the laser pulse.

Analytical modeling of a device for vibration isolation and energy harvesting using simply-supported bi-stable hybrid symmetric laminate

The bi-stable hybrid symmetric laminate (BHSL) has been investigated for vibration isolation and energy harvesting, by taking advantage of its bi-stable characteristic. A device using simply-supported BHSL is proposed to integrate these two functions. This device is called the quasi-zero-stiffness (QZS) vibration isolator with a piezoelectric element (PE). An equivalent theoretical model is developed based on the restoring force-displacement curve of BHSL. The approximate analytical solutions and direct numerical solutions are obtained by the harmonic balance method and the Runge-Kutta method, respectively. Displacement transmissibility is utilized to quantify the vibration isolation performance of this vibration isolator. The influences of the excitation amplitude, the length of BHSL, the width of BHSL and the load resistance are analyzed. The results show that adjusting these parameters can improve the vibration isolation performance and energy harvesting performance. The QZS vibration isolator with PE achieves the goal of higher power density under a low-frequency and low-amplitude excitation and simultaneously has a good vibration isolation performance.

Statistical study on the derailment factor considering the weak correlation of lateral-vertical wheel-rail forces of a vehicle-bridge system

This paper proposes a stochastic method to solve the weak correlation of lateral-vertical forces when the train crosses a bridge. This method treats the Kalker creep equation with Taylor series and uses the moment statistical method to approximate the numerical characteristics of the nonlinear terms of the equation. The variance recursive method, mean recursive method, and lateral linear creep force modulation function are proposed. Based on these methods, the approximate numerical solutions for the standard deviation of the lateral linear creep force and the correlation coefficient of the lateral-vertical forces are obtained and verified by Monte Carlo method (MCM). Under certain preconditions, the derailment factor considering the correlation of the wheel-rail forces obeys the Approximate Cauchy Distribution, which is verified by the Kolmogorov–Smirnov test. The correctness of the proposed method is validated comparing the similarity of the derailment factor extreme value cumulative distribution function curve obtained by MCM. The case of a 6-car Pioneer EMU running through a 3-span 32 m simply-supported bridge is concerned by the proposed method and it is found that there is a significant difference with different traffic intense of the railway.

SOLUTION TO THE KDV EQUATION USING FEM

The main objective of the following document is to present an approximate numerical solution of the Korteweg de Vries (KdV) equation, since it has a wide field of applications, such as acoustics, molecular biology, optics, and quantum mechanics. For this, the basic concepts associated with the numerical method that will be used to obtain the approximate solution will be presented initially, in our case, the finite element method and in particular the Galerkin method. A further understanding of the effectiveness of the method is achieved by presenting an analytical solution considering the wave speed as a constant and performing a change of variable. Subsequently, the approximate numerical solution for different times is compared with the analytical solution obtained. As a secondary objective, we propose different simulations corresponding to the analytical solution for different times, the approximate solution through the Galerkin method to observe the behavior of the displacement of a particle on the wave in one dimension. Keywords: Approximate Solution, Finite Element Method, The Galerkin Solution, KdV Equation, Nonlinear Equation DOI： https://doi.org/10.35741/issn.0258-2724.58.3.34

Primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force

<abstract><p>In the present paper, the primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force is studied. The approximately analytical solution and the amplitude-frequency equation are obtained using the multiple scale method. Based on the Lyapunov theory, the stability conditions for the steady-state solution are obtained. The bifurcations of primary resonance for system parameters are analyzed, and the influence of parameters on fractional-order model is also studied. Numerical simulation shows that when the parameter values are fixed, the curve bends to the right or left, resulting in jumping phenomena and multi-valued amplitudes. As the excitation frequency changes, the typical hardening or softening characteristics of the oscillator are observed. In addition, the comparisons of approximate analytical solution and numerical solution are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution.</p></abstract>

An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator

<abstract><p>In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.</p></abstract>

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval