Approximate Numerical Solutions Research Articles

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for 0⩽ρ⩽12 when σ = 10 and β=8/3 , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

The impact of rotation on the onset of cellular convective movement in a casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations

In this effort, we examined the impact of rotation on the arrival of cellular convective motion in a Casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations. The problem is important to cellular foams prepared from plastics, ceramics, and metallic where radiation conductivity is revealed as a power function of temperature. The altered Darcy model is used to characterize the rheological performance of Casson fluid flow in permeable medium. The approximate analytical solution and numerical solution correct to one decimal place are presented utilizing the Galerkin method. The analysis reveals that the influence of thermal conductivity disparity parameter and the rotation is to delay the convective motion whereas; the viscosity disparity parameter and the Casson parameter have dual impact on the convective motion in the presence of rotation. The range of the convective cell drops with increasing the thermal conductivity disparity parameter, the viscosity disparity parameter, the Casson parameter and rotation parameter. In the absence of rotation, the range of the convective cell does not depend on the Casson parameter and the viscosity disparity parameter. Further, the existing results are compared with the existing literature under the particular case of this study.

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Honeycomb-configured dissipative nanofluid flow within a squeezed channel with entropy generation: regression and numerical evaluations

PurposeNanosized honeycomb-configured materials are used in modern technology, thermal science and chemical engineering due to their high ultra thermic relevance. This study aims to scrutinize the heat transmission features of magnetohydrodynamic (MHD) honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts.Design/methodology/approachMass, energy and momentum preservation laws are assumed to find the mathematical model. A set of unified ordinary differential equations with nonlinear behavior is used to express the correlated partial differential equations of the established models, adopting a reasonable similarity adjustment. An approximate convergent numerical solution to these equations is evaluated by the shooting scheme with the Runge–Kutta–Fehlberg (RKF45) technique.FindingsThe impression of pertinent evolving parameters on the temperature, fluid velocity, entropy generation, skin friction coefficients and the heat transference rate is explored. Further, the significance of the irreversibility nature of heat transfer due to evolving flow parameters are evaluated. It is noted that the heat transference rate performance is improved due to the imposition of the allied magnetic field, Joule dissipation, heat absorption, squeezing and thermal buoyancy parameters. The entropy generation upsurges due to rising magnetic field strength while its intensification is declined by enhancing the porosity parameter.Originality/valueThe uniqueness of this research work is the numerical evaluation of MHD honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts. Furthermore, regression models are devised to forecast the correlation between the rate of thermal heat transmission and persistent flow parameters.

Variable order fractional diabetes models: numerical treatment

ABSTRACT In this paper, we present two numerical approximation solutions for the updated nonlinear variable order fractional differential equations of diabetes disease. The variable order fractional differential operator is based on the Caputo fractional derivative concept. The first used method is developed using the nonstandard finite difference process, such that a new quick approximation is applied to tackle the nonlinear terms. By applying some reasonable assumptions to the given data, we can prove that this scheme maintains the positivity and boundedness of the discretized solutions. The second method is based on approximation with shifted Jacobi polynomials. The properties of Jacobi polynomials, together with the shifted Jacobi-Gauss-Lobatto nodes were utilized to reduce the proposed system into a set of algebraic nonlinear equations. Furthermore, some numerical experiments are conducted to compare the performance of the introduced schemes. Our numerical simulations show that the nonstandard finite difference approach requires less computational time compared to the Jacobi-Gauss-Lobatto spectral collocation. While the last one is more accurate when used for solving the variable order fractional diabetes model.

A new method for determining crack propagation

This paper presents a new method for obtaining approximate closed-form solutions for crack propagation problems. In general, it is not possible to obtain explicit solutions of the parameters and elementary functions of these problems, due to their terms complexity, present in their definitions. In this way, we used numerical methods to obtain approximate numerical solutions. The method presented in this article provides a proposal, which, when defined by differential equations, is applicable and extensible to other problems. The approximate solutions are obtained as an exact solution of a modified initial value problem. The performance is evaluated in relation to problems with known exact solution.

Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management.

A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

Abstract In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Numerical method of the exact control for the elastic string problem with moving boundary

The objective of this article is to investigate, analytically and numerically, the exact control problem associated with a mathematical model that describes the vertical vibrations of elastic strings, with moving contours. To determine the approximate numerical solution, following the description of the HUM (Hilbert Uniqueness Method), the finite element method associated with the Newmark method was used for the adjunct problem and the backward problem. To calculate the optimal control, the conjugated gradient method is used. The properties of a numerical problem must be consistent with the theoretical mathematical one. However, in order to obtain the numerical control, the grid is of fundamental importance, so we used the Bi-Grid technique. In this way, we developed an algorithm to solve this problem. Numerical simulations were performed for different initial data and moving contours, with and without the Bi-Grid algorithm and the numerical results confirmed the consistency between the theoretical and numerical results.

Fuzzy Finite Elements Solution Describing Recession Flow in Unconfined Aquifers

In this work, a novel fuzzy FEM (Finite Elements Method) numerical solution describing the recession flow in unconfined aquifers is proposed. In general, recession flow and drainage problems can be described by the nonlinear Boussinesq equation, while the introduced hydraulic parameters (Conductivity K and Porosity S) present significant uncertainties for various reasons (e.g., spatial distribution, human errors, etc.). Considering the general lack of in situ measurements for these parameters as well as the certain spatial variability that they present in field scales, a fuzzy approach was adopted to include the problem uncertainties and cover the disadvantage of ground truth missing data. The overall problem is encountered with a new approximate fuzzy FEM numerical solution, leading to a system of crisp boundary value problems. To prove the validity and efficiency of the new fuzzy FEM, a comparative analysis between the proposed approach and other well-known and tested approximations was carried out. According to the results, the proposed FEM numerical solution agrees with Karadinumerical method for the crisp case and is in close agreement with the original analytical solution proposed by Boussinesq in 1904 with the absolute reduced error to be 4.6‰. Additionally, the possibility theory is applied, enabling the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of hydraulic properties (e.g., water level, outflow volume) and make the right decisions for rational and productive engineering studies.

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines

Abstract This article develops approximate numerical solutions through the generalized method of lines for the time-independent, incompressible Navier–Stokes system in fluid mechanics. More specifically, we highlight the main objective of this article is the development of new approximate procedures for solving numerically the equation systems originated from a domain discretization related to a finite differences scheme. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines (or more generally in curves), and the concerning solutions are written on these lines as functions of the boundary conditions and the domain boundary shape. Finally, it is worth emphasizing that in this text, we have presented softwares and results for a concerning approximate proximal approach, as well as results based on the original conception of the generalized method of lines.

WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modeled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modeling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (2018) [3] to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.

TOUCHARD POLYNOMIALS APPROACH FOR SOLVING NON- LINEAR FREDHOLM INTEGRO- DIFFERENTIAL EQUATION OF SECOND TYPE

A numerical approach to solve Non- linear Fredholm integro- differential (NLFID) equation of the 1st order and 2nd type has been proposed. The approach was based upon Touchard polynomials. The non-linear Fredholm integro- differential equation was changed into a system of non- linear algebraic equations and solved using Newton repeating approach. The proposed approach was evaluated by displaying three numerical problems, and the approximate numerical solutions were compared with exact solution and four methods in the literature. MATLAB R2018b was used to perform all calculations and graphs.

Stability analysis and numerical approximate solution for a new epidemic model with the vaccination strategy

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible‐High risk‐Infective‐Recovered‐Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, . It is proved that the disease‐free equilibrium is asymptotically stable if , and it is unstable if . Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre‐Ritz‐Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high‐risk people for vaccination can prevent them from getting infected and reduce mortality in the community.

A Finite Difference Technique for Numerical Solution of the Boundary Value Problem in ODEs of Order Three

In the article, we study the approximate numerical solution to the boundary value problem in ordinary differential equations. In the present article, a third-order two-point boundary value problem is considered for discussion. We developed a second order accurate finite difference method for the approximate numerical solution of the considered problem. We took a special boundary condition; we did not find this boundary condition in the literature. We have discussed the standard convergence analysis of the proposed method. Numerical experiments on linear, nonlinear, and obstacle problems approve the order of accuracy and efficiency of the method.

Numerical method for a compound Poisson risk model with liquid reserves and proportional investment

<abstract><p>In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.</p></abstract>

LOCAL TIME FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING LOCAL TIME FRACTIONAL TELEGRAPH EQUATIONS

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when [Formula: see text]. It also required minimal amount of computational work and saved a lot of time.

How to achieve carbon abatement in aviation with hybrid mechanism? A stochastic evolutionary game model

Aviation carbon emissions are growing as the volume of aviation traffic continues to increase, both exacerbating global greenhouse gas emissions and increasing the resistance to green aviation development. To effectively promote aviation carbon abatement, this paper, based on the hybrid mechanism composed of carbon trading and taxation, a tripartite Itô stochastic evolutionary game model is first constructed. The complex game interactions among the administration, major airlines, and minor airlines are analyzed. Second, we use Gaussian white noise as the ambient uncertainty and apply stochastic Taylor expansion to find the numerical approximation solution. Finally, through numerical simulations, the decision-making behavior of stakeholders and their sensitivity to key influencing factors are illustrated. The study shows that different variables have differential effects on stakeholders’ strategic choices in terms of convergence speed, change speed, and stability. Starting from three different regulatory paths, this study provides insights into the priority and direction of adjusting relevant variables, thereby offering guidance for policymakers and managers in effectively regulating aviation carbon abatement.