Showing papers on "Numerical analysis published in 2022"

Computational study on the dynamics of fractional order differential equations with applications

Abstract: In this research work, the analysis of general fractional order system is investigated under Atangana, Baleanu and Caputo (ABC) fractional order derivative. Our study is related to three aspects including existence theory, stability and numerical analysis. For existence theory, we use Krasnoselskii and Banach contraction theorems. Further using nonlinear analysis, we develop some necessary results for Ulam Hyer’s (UH) stability. The approximate solution is computed by using Adam’s-Bashforth numerical technique. For justification, we provide three concert examples along with necessary numerical and graphical interpretations.

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Abstract: The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

On spectral numerical method for variable-order partial differential equations

Abstract: In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.

Fractional boundary element solution of three-temperature thermoelectric problems

Abstract: Abstract The primary goal of this article is to propose a new fractional boundary element technique for solving nonlinear three-temperature (3 T) thermoelectric problems. Analytical solution of the current problem is extremely difficult to obtain. To overcome this difficulty, a new numerical technique must be developed to solve such problem. As a result, we propose a novel fractional boundary element method (BEM) to solve the governing equations of our considered problem. Because of the advantages of the BEM solution, such as the ability to treat problems with complicated geometries that were difficult to solve using previous numerical methods, and the fact that the internal domain does not need to be discretized. As a result, the BEM can be used in a wide variety of thermoelectric applications. The numerical results show the effects of the magnetic field and the graded parameter on thermal stresses. The numerical results also validate the validity and accuracy of the proposed technique.

A finite point method for the fractional cable equation using meshless smoothed gradients

Abstract: This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.

Fractional boundary element solution of three-temperature thermoelectric problems

Abstract: Abstract The primary goal of this article is to propose a new fractional boundary element technique for solving nonlinear three-temperature (3 T) thermoelectric problems. Analytical solution of the current problem is extremely difficult to obtain. To overcome this difficulty, a new numerical technique must be developed to solve such problem. As a result, we propose a novel fractional boundary element method (BEM) to solve the governing equations of our considered problem. Because of the advantages of the BEM solution, such as the ability to treat problems with complicated geometries that were difficult to solve using previous numerical methods, and the fact that the internal domain does not need to be discretized. As a result, the BEM can be used in a wide variety of thermoelectric applications. The numerical results show the effects of the magnetic field and the graded parameter on thermal stresses. The numerical results also validate the validity and accuracy of the proposed technique.

Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis

Abstract: In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.

Essence of MIKE 21C (FDM Numerical Scheme): Application on the River Morphology of Bangladesh

Abstract: Understanding, quantifying, and forecasting water flow and its behavior in environment is made possible by the use of computational hydraulics in con-junction with numerical models, which is one of the most powerful tools currently available. It is made up of simple to complex mathematical equations having linear and/or nonlinear elements, as well as ordinary and partial differential equations, and it is used to solve problems in many areas. In the vast majority of cases, it is not useful to reach analytical solutions to these mathematical equations using conventional methods. In these settings, mathematical models are solved by employing a variety of numerical algorithms and associated schemes. As a result, in this manuscript, we will cover the most fundamental numerical approach, the Finite Difference Method (FDM), in order to reformulate the governing equations for water and sediment flow from a system of partial differential equations to a system of linear equations. As part of our analysis into the inner workings of a computer program known as MIKE 21C, we will attempt to gain a better understanding of the hydrodynamic processes that take place in major rivers in Bangladesh. In addition to that, we will go over some of the most commonly used morphological studies that have been conducted on Bangladesh’s major rivers, including morphological solutions that have been developed in response to water supply con-cerns.

Convergence Analysis of the Variational Operator Splitting Scheme for a Reaction-Diffusion System with Detailed Balance

Abstract: We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms. In addition, a combination of rough error estimate and refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting procedure results in the convergence estimate of the numerical scheme for the full reaction-diffusion system. The convergence analysis technique could be extended to a more general class of dissipative reaction mechanisms. As an example, we also consider a near-equilibrium reaction kinetics, which was derived by the linear response assumption on the reaction trajectory. Although the reaction rate is more complicated in terms of concentration variables, we show that the numerical approach and the convergence analysis also work in this case.

Sensitivity of Compressed Composite Channel Columns to Eccentric Loading

Abstract: This study investigated short thin-walled channel columns made of carbon/epoxy laminate. Columns with two multi-ply composite layups [0/45/−45/90]s and [90/−45/45/0]s were tested, with each layup having eight plies symmetric to the midplane. The columns were subjected to compressive loads, including an eccentric compressive load applied relative to the center of gravity of their cross-section. Simple support boundary conditions were applied to the ends of the columns. The scope of the study included analyzing the effect of load eccentricity on the buckling mode, bifurcation load (idealized structure), and critical load (structure with initial imperfections). The critical load for the actual structure was determined with the use of approximation methods, based on experimental postbuckling equilibrium paths. In parallel with the experiments, a numerical analysis was conducted using the finite element method and Abaqus® software (Dassault Systèmes, Vélizy-Villacoublay, France). The first stage of the numerical analysis consisted of solving an eigenproblem, in order to determine the mode of the loss of structural stability and to calculate the bifurcation loads for structures under axial and eccentric compression. The second stage of the numerical analysis involved examining the non-linear state of pre-deflected structures. Numerical postbuckling equilibrium paths were used to estimate the critical loads with an approximation method. The experimental results were used to validate the numerical models. This made it possible to determine the effect of compressive load eccentricity on the buckling mode and critical load of the tested structures. The results confirmed that compressive load eccentricity had a significant impact on the load-carrying capacity in the postbuckling range. This may potentially lead to premature damage to composite materials and, ultimately, to a reduced load-carrying capacity of structures.