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The novelty of this contribution is to propose an implicit numerical scheme for solving time-dependent boundary layer problems. The scheme is multi-step and consists of two stages. It is third-order accurate in time and constructed on three-time levels. For spatial discretization, a fourth-order compact scheme is adopted. The stability of the proposed scheme is analyzed for scalar linear partial differential equation (PDE) that shows its conditional stability. The convergence of the scheme is also provided for a system of time-dependent parabolic equations. Moreover, a mathematical model for heat and mass transfer of mixed convective Williamson nanofluid flow over flat and oscillatory sheets is modified with the characteristic of the Darcy–Forchheimer model. The results show that the temperature profile rises by developing thermophoresis and Brownian motion parameter values. Also, the proposed scheme is compared with an existing Crank–Nicolson method. It is found that the proposed scheme converges faster than the existing one for solving scalar linear PDE as well as the system of linear and nonlinear parabolic equations, which are dimensionless forms of governing equations of a flow phenomenon. The findings provided in this study can serve as a helpful guide for future investigations into fluid flow in closed-off industrial settings. [ABSTRACT FROM AUTHOR]

In this article, a new numerical algorithm for solving a 1-dimensional (1D) and 2-dimensional (2D) time-fractional diffusion equation is proposed. The Sinc-Galerkin scheme is considered for spatial discretization, and a higher-order finite difference formula is considered for temporal discretization. The convergence behavior of the methods is analyzed, and the error bounds are provided. The main objective of this paper is to propose the error bounds for 2D problems by using the Sinc-Galerkin method. The proposed method in terms of convergence is studied by using the characteristics of the Sinc function in detail with optimal rates of exponential convergence for 2D problems. Some numerical experiments validate the theoretical results and present the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze the alternating direction implicit (ADI) difference schemes in conjunction with the second order backward differentiation formula (BDF2) method with variable time step sizes for solving the two-dimensional parabolic equation. The spatial compact operators are also applied to construct high order ADI scheme. By using the discrete energy method and the positive definiteness of the nonuniform BDF2 approximation, we prove the unconditional H 1 semi-norm stability for the variable-step BDF2-ADI scheme and the variable-step compact BDF2-ADI scheme under the constraint r n ≤ 4.8 , where r n denotes the adjacent step size ratio. Moreover, the optimal second order and the fourth order convergence rates are derived rigorously under this restriction. To the best of our knowledge, this is the first strict theoretical analysis of the variable-step ADI numerical schemes for the multidimensional parabolic equations. Several numerical examples are included to verify the analysis results. The extension to the nonlinear Allen-Cahn equation is also presented, for which the variable-step BDF2-ADI scheme and the corresponding compact scheme combined with the adaptive time-stepping algorithm improve the efficiency of the long time simulation. [ABSTRACT FROM AUTHOR]

This article compared two high-order numerical schemes for convection-diffusion delay differential equation via two fractional operators with singular kernels. The objective is to present two effective schemes that give (3 - α) and second order of accuracy in the time direction when α ∈ (0 , 1) using Caputo and Modified Atangana-Baleanu Caputo derivatives, respectively. We also implemented a trigonometric spline technique in the space direction, giving second order of accuracy. Moreover, meticulous analysis shows these numerical schemes to be unconditionally stable and convergent. The efficiency and reliability of these schemes are illustrated by numerical experiments. The tabulated results obtained from test examples have also shown the comparison of these operators. [ABSTRACT FROM AUTHOR]

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields. [ABSTRACT FROM AUTHOR]

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald difference operator and product trapezoidal formula were utilized to construct a high-order numerical scheme. Also, we analyzed the stability and convergence. The convergence order was O (τ 2 + h x 4 + h y 4) , where τ is the time step size, h x and h y are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning. [ABSTRACT FROM AUTHOR]

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2 − α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

A neural network model is constructed to solve convex quadratic multi-objective programming problem (CQMPP). The CQMPP is first converted into an equivalent single-objective convex quadratic programming problem by the mean of the weighted sum method, where the Pareto optimal solution (POS) are given by diversifying values of weights. Then, for given various values weights, multiple projection neural networks are employded to search for Pareto optimal solutions. Based on employing Lyapunov theory, the proposed neural network approach is established to be stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the single-objective problem. The simulation results also show that the presented model is feasible and efficient. [ABSTRACT FROM AUTHOR]

We consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the L2$L^2$ and the H1$H^1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a courant‐friedrichs‐lewy (CFL) condition between the space mesh size and the time step sizes. [ABSTRACT FROM AUTHOR]

To describe non-continuous control strategy triggered by a predefined ratio of the concerned population in disease progression, an SIR epidemic model with non-monotone incidence is established, dealing with the integrated interventions of vaccination and isolation as ratio-dependent impulsive control. Based on the analysis of multiple equilibria and stability for the continuous subsystem, complex property of the impulsive controlled system is conducted theoretically and numerically. It is found that the ratio magnitude as well as the initial value plays an important role in the threshold dynamics, determining the trajectory convergence to the order-k(k < 3 ) periodic solution, the endemic equilibria or the disease free equilibrium under certain conditions. Moreover, numerical examples are provided to demonstrate the theoretical results, and in particular, the case of converging to an order-2 periodic solution is presented in simulation, revealing the influence of the ratio threshold control on disease transmission. [ABSTRACT FROM AUTHOR]

This paper explores the possibility of solving the one-dimensional Burgers' equation with a tailored finite point method (TFPM) based on an explicit stencil along with utilizing the Cole Hopf transformation. The proposed TFPM procedure operates on an explicit four-point stencil where the nodal solution at the advanced temporal level is written as a linear combination of the nodal solutions at the preceding temporal level. In order to bring the essence of the local exact solutions into the numerical approximations the scalar coefficients involved in the linear sum are determined by the application of the fundamental solutions into the stencil. The efficiency of the method is demonstrated through the comparisons of TFPM solutions for various examples with the exact solutions and solutions from well-established methodologies in the literature. [ABSTRACT FROM AUTHOR]

Summary: The convergence of filtered error and filtered reference adaptive feedforward is limited by three effects: model mismatch, unintended input‐disturbance interaction and too fast parameter adaptation. In this article, the first two effects are considered for MIMO systems under the slow parameter adaptation assumption. The convergence with model mismatch is conventionally guaranteed using a strictly positive‐real condition. This condition can be easily verified in the frequency domain, but due the high‐frequency parasitic dynamics of real systems, it is hardly ever satisfied. Nevertheless, filtered error and filtered reference adaptive feedforward have successfully been implemented in numerous applications without satisfying the strictly positive‐real condition. It is shown in this article that the strictly positive‐real condition can be relaxed to a power‐weighted integral condition, that is less conservative and provides a practical check for the convergence of filtered error adaptive feedforward for real systems in the frequency domain. The effects of input‐disturbance interaction are analysed and conditions for the stability are given in the frequency domain. Both conditions give clear indicators for frequency domain filter tuning, and are verified on an experimental active vibration isolation system. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Constructing a higher order collocation framework for solving the Caputo multi-term time-fractional advection–diffusion and diffusion-type problems is the primary objective of this work, which has influenced the field of scientific disciplines. Advection–diffusion and reaction–diffusion equations were developed by modeling scientific phenomena in fluid flow issues, solid oxide fuel cells, and solvent diffusion into heavy oils. As a result, numerical solutions to these problems have garnered significant attention. The L 1 - 2 approximation approach approximates the fractional derivatives of orders η , η i ∈ (0 , 1) that are present in the considered problem. This approach provides a higher accuracy of O (k 3 - max { η , η i }) in time direction. Fourth-order convergence in space is achieved by employing a spline collocation technique with trigonometric quintic splines. Results from applying the suggested computational approach to four test examples have demonstrated its superiority and validity. [ABSTRACT FROM AUTHOR]

Abstract In this article, a new numerical algorithm for solving a 1-dimensional (1D) and 2-dimensional (2D) time-fractional diffusion equation is proposed. The Sinc-Galerkin scheme is considered for spatial discretization, and a higher-order finite difference formula is considered for temporal discretization. The convergence behavior of the methods is analyzed, and the error bounds are provided. The main objective of this paper is to propose the error bounds for 2D problems by using the Sinc-Galerkin method. The proposed method in terms of convergence is studied by using the characteristics of the Sinc function in detail with optimal rates of exponential convergence for 2D problems. Some numerical experiments validate the theoretical results and present the efficiency of the proposed schemes.

Scientific inquiry into effective numerical methods for modelling complex physical processes has led to the investigation of fluid dynamics, mainly when non-Newtonian properties and complex heat sources are involved. This paper presents an enhanced exponential time integrator approach to dynamically simulate non-Newtonian boundary layer flow with spatially and temporally varying heat sources. We propose an explicit scheme with second-order accuracy in time, demonstrated to be stable through Fourier series analysis, for solving time-dependent partial differential equations (PDEs). Utilizing this scheme, we construct and solve dimensionless PDEs representing the flow of Williamson fluid under the influence of space- and temperature-dependent heat sources. The scheme discretizes the continuity equation of incompressible fluid and Navier–Stokes, energy, and concentration equations using the central difference in space. Our analysis illuminates how factors affect velocity, temperature, and concentration profiles. Specifically, we observe a rise in temperature profile with enhanced coefficients of space and temperature terms in the heat source. Non-Newtonian behaviours and geographical/temporal variations in heat sources are critical factors influencing overall dynamics. The novelty of our work lies in developing an explicit exponential integrator approach, offering stability and second-order accuracy, for solving time-dependent PDEs in non-Newtonian boundary layer flow with variable heat sources. Our results provide valuable quantitative insights for understanding and controlling complex fluid dynamics phenomena. By addressing these challenges, our study advances numerical techniques for modelling real-world systems with implications for various engineering and scientific applications.

The paper demonstrates two approaches to constructing monotonic difference schemes for the transfer equation in divergent form from the family of semi-Lagrangian methods: Eulerian–Lagrangian and Lagrangian–Eulerian. Within each approach, a monotonic conservative difference scheme is proposed. It is shown that within the framework of the Lagrangian–Eulerian approach, based on the use of curvilinear grids formed by the characteristics of the approximated transfer operator, it is possible to construct monotonic difference schemes of second order accuracy. [ABSTRACT FROM AUTHOR]

Summary: In the presented study, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is calculated using a recurrent neural network (RNN) model. Since there is a few research for solving FNLOP by RNN's, we give a new approach to solve the problem. By reducing the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are given. Moreover, we use the KKT conditions into a RNN as an important tool to solve the problem. Besides, the global convergence properties and the Lyapunov stability of the dynamic model are studied in this study. In the final step, some illustrative examples are considered to establish the obtained results. Reported results are compared with some others network models. [ABSTRACT FROM AUTHOR]

A neural network architecture is constructed to solve Convex Multi-objective Programming Problem (CMPP). By using the weighted sum method, a single-objective optimization problem related to the CMPP is formulated, in which the scalar objective is summation of weighted original objective functions. The Pareto Optimal Solution (POS) is given by using different values of weights. A neural network framework is then designed for solving the obtained convex problem. Based on employing Lyapunov theory, the proposed model is established to be stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the convex programming problem with different values of weights. The computer simulations shows the feasibility and the efficiency of the suggested model. [ABSTRACT FROM AUTHOR]

In the proposed manuscript, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is gained using a projection recurrent neural network (RNN) scheme. Since there is a few research on resolving FNLOP by RNNs, we establish a new scheme to solve the problem. By reducing the original program to an interval problem and then a weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are presented. Moreover, we apply the KKT conditions to a RNN as an efficient tool to solve the problem. Besides, the convergence properties and stability analysis of the system model are provided. In the final step, several simulation examples are verified to support the obtained results. Reported results are compared with some other previous neural networks. [ABSTRACT FROM AUTHOR]

Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit, or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta--Nyström method. [ABSTRACT FROM AUTHOR]

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid's fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme. [ABSTRACT FROM AUTHOR]

An efficient numerical method with high accuracy both in time and in space is proposed for solving d$$ d $$‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an s$$ s $$‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order s+1$$ s+1 $$ in time, when the implicit Runge‐Kutta method with classical order p$$ p $$ (p≥s+1$$ p\ge s+1 $$) is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter α$$ \alpha $$. This improves the previous result which depends on the fractional parameter α$$ \alpha $$. Numerical experiments verify and complement our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, two difference schemes are proposed to solve numerically a nonlinear strong damped wave equation in two dimensions. The first scheme is nonlinear implicit, the second scheme is linear implicit. It is proved that the two schemes are unique solvable, unconditionally stable. Moreover, the present numerical methods are convergent with second order accuracy both in space and time in the discrete H 1 -norm. Numerical experiments are provided to show the validity and the accuracy of the proposed difference schemes as well as their efficiency. [ABSTRACT FROM AUTHOR]

Nonlinear Kawarada equations have been used to model solid fuel combustion processes in the oil industry. An effective way to approximate solutions of such equations is to take advantage of the finite difference configurations. Traditionally, the nonlinear term of the equation is linearized while the numerical stability of a difference scheme is investigated. This leaves certain ambiguity and uncertainty in the analysis. Based on nonuniform grids generated through a quenching-seeking moving mesh method in space and adaptation in time, this paper introduces a completely new stability analysis of the approximation without freezing the nonlinearity involved. Pointwise orders of convergence are investigated numerically. Simulation experiments are carried out to accompany the mathematical analysis to strengthen our conclusions. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm for both u$$ u $$ and ρ$$ \rho $$. Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

Iterative learning control (ILC) algorithms appeared in connection with the problems of increasing the accuracy of performing repetitive operations by robots. They use information from previous repetitions to adjust the control signal on the current repetition. Most often, information from the previous repetition only is used. ILC algorithms that use information from several previous iterations are called higher-order algorithms. Recently, interest in these algorithms has increased in the literature in connection with robotic additive manufacturing problems. However, in addition to the fact that these algorithms have been little studied, there are conflicting estimates regarding their properties. This paper proposes new higher-order ILC algorithms for linear discrete and differential systems. The idea of these algorithms is based on an analogy with multi-step methods in optimization theory, in particular, with the heavy ball method. An example is given that confirms the possibility to accelerate convergence of the learning error when using such algorithms. [ABSTRACT FROM AUTHOR]

In this paper, the development and analysis of a proposed method for solving physical models arising from real-life scenarios are presented. The proposed method is derived via the transcendental function of exponential type, examined and studied for its properties. In addition, the effectiveness of the method is evaluated by applying it to three numerical examples that originated from real-world scenarios. Moreover, this study presents a comparison of the outcomes generated by the proposed method and the existing method, in the context of the exact solution. The study concludes that the proposed method solves real life problems with the expected level of accuracy and, therefore, can be considered among the numerous methods that are appropriate and suitable for solving first-order initial value problems (IVPs). [ABSTRACT FROM AUTHOR]

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes. • This work presents a minimal machinery algorithm on an operator splitting technique in conjunction with a tailored FPM for a 2D Burgers' equation. • The tailored FPM is advisable for its efficient adaptation to local solution properties, user-friendly algorithm, and cost-effectiveness. • The manuscript offers a comprehensive mathematical and computational background along with stability, consistency, and convergence analyses. • The algorithm despite being simple and straightforward has performed on a par with other methods based on elite classical formulation techniques. [ABSTRACT FROM AUTHOR]

This paper considers an autonomous system of singularly perturbed equations of fast variables, consisting of first-order equations and one equation of a slow variable. The first approximation matrix of singularly perturbed equations has pairwise complex conjugate eigenvalues. The system has an equilibrium position, and the stability of the equilibrium position is lost by all eigenvalues at some value of the slow variable. It is proven that the solution of a singularly perturbed equation remains near an unstable equilibrium position during a finite time. Thus, the solution is delayed near the unstable equilibrium position. Early works considered cases when the stability of the equilibrium position is lost by one pair of complex conjugate eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, an implicit finite difference scheme combined with Kansa meshless method is proposed for the two-dimensional time and space fractional Bloch-Torrey equation. The Caputo derivative with respect to time is discretized by finite difference method based on Alikhanov's super convergent approximation while Riesz derivative with respect to spatial variables by Kansa meshless method. The convergence and stability of the time semi-discretization are proven using energy method. The convergence order in time direction is proven to be 2. The implementation of the full discretization is discussed in detail. Especially, the issue of calculation of Riesz derivative of RBF is addressed. Numerical examples are given which verify the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations' precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids. [ABSTRACT FROM AUTHOR]

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for one- and two-dimensional Riesz space variable-order fractional diffusion equations. We first introduce fractional centered difference (FCD) and weighted and shifted fractional centered difference (WSFCD) schemes for Riesz space variable-order fractional derivatives. Then the Crank-Nicolson (CN) scheme and the linearly implicit conservative (LIC) difference scheme are applied to discretize the time derivative in linear and nonlinear problems, respectively. Thus, we get CN-FCD and CN-WSFCD schemes, and LIC-FCD and LIC-WSFCD schemes, respectively. Theoretical results about the stability and convergence for the above-mentioned schemes are presented and proved. Banded preconditioners are introduced to speed up GMRES methods for solving the discretization linear systems. The spectral property of the preconditioned matrix is analyzed. Numerical results show that the proposed schemes and preconditioners are very efficient. [ABSTRACT FROM AUTHOR]

In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for the existence and uniqueness, as well as the stability of scheme, along with a priori error estimates. Finally, we validate our theoretical results through various numerical computations. [ABSTRACT FROM AUTHOR]

This paper aims in developing the finite difference scheme for riemann liouville space fractional diffusion equation and caputo fabrizio time fractional derivative with a Dirichlet BCs. The prove for scheme to be unconditionally stable and also convergent is been discussed, and the convergence order of our scheme is O Δ t 2 + h 2 . Further, an application example in terms of numerical results is solved and graphs simulated using Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a new numerical approximation for the Caputo fractional derivative and apply it for the 1D case to obtain a time-stepping method. We then develop an alternating direction implicit (ADI) scheme for the 2D case. Using the discrete energy method, we prove that the proposed difference schemes are unconditionally stable and convergent in both 1D and 2D cases. Finally, several numerical examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this article is proposed a method for numerically solving a mixed problem for a hyperbolic equation with memory. An implicit difference scheme is constructed as an effective means for solving complex multidimensional problems of mathematical physics. A condition is found for guaranteed stability of an implicit difference scheme for a mixed problem in the -norm. The order of convergence for all variables of the presented difference scheme was calculated and confirmed by numerical experiment. [ABSTRACT FROM AUTHOR]

Bipolar-fuzzy differential equations applied for modeling in science, engineering, and social science. The aim of this paper is to provide a numerical method to determine the preferred level of accuracy with less computational cost. The proposed method is generated by combining three-step explicit and two-step implicit bipolar-fuzzy methods, which are obtained by the spline interpolation. For more illustration, the consistency, stability, and convergence theorems are discussed in detail. Finally, for show the performance and efficiency of the method, several numerical examples are presented. [ABSTRACT FROM AUTHOR]

Mathematical models and numerical simulations have been prominently successful in providing insight for the exploration of evolution mechanism in scientific engineering. In this paper, a linearized fourth-order compact alternating direction implicit (ADI) method is developed to numerically solve a mathematical model, which includes phytoplankton–zooplankton interactions arising in marine ecosystem. The solvability, convergence and stability of the proposed method are discussed as well. It is proven that the proposed method is fourth-order accurate in space and second-order accurate in time. Finally, we present several numerical simulations to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes. [ABSTRACT FROM AUTHOR]

AbstractThis study introduces a novel and versatile fractal finite difference scheme designed to address unsteady flow challenges in quantum calculus over flat and oscillatory sheets. Motivated by the need to advance our understanding of nanofluid dynamics, particularly in heat and mass transportation, this research aims to bridge existing gaps and contribute to improving computational methodologies. Our approach involves a two-stage scheme incorporating explicit and implicit methods, focusing on q-derivatives for spatial terms within the mathematical model. The system encompasses the Navier-Stokes equation, energy equation, and concentration equation, offering a comprehensive framework for analyzing mixed convective Williamson nanofluid flow under the influence of chemical reactions. The convergence condition for the time-dependent partial differential equations system, particularly the scalar convection-diffusion equation, is presented. Stability requirements for our proposed fractal scheme are also delineated. Notably, our study explores the implications of the Weisenberg parameter on the velocity profile, revealing a decline that underscores its influence on nanofluid behavior. Our fractal finite difference scheme demonstrates a substantially faster convergence rate than the fractal Crank-Nicolson scheme. This outcome emphasizes the computational efficiency and efficacy of our proposed technique. In a broader context, our work contributes to the field by leveraging fractal geometry and quantum calculus to enhance our understanding of nanofluid dynamics. These tools present exciting opportunities for groundbreaking applications and technological advancements across various industries. As we delve into these research areas, our work sets the stage for exploration, providing a framework with great potential for transformative progress in nanofluid dynamics. [ABSTRACT FROM AUTHOR]

Relying upon Legendre polynomials, an improved multiwavelet orthonormal basis is constructed in the reproducing kernel space. We study boundary value problems (BVPs) based on the improved wavelet orthonormal basis and construct the ε$$ \varepsilon $$‐best approximate solution. Convergence and the stability are discussed; this method has higher order convergence. The main advantage of the wavelet based our method is its efficiency and simple applicability for a variety of boundary conditions. Some numerical tests even including singular BVPs reveal the stability and efficiency of the new algorithm. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present a finite difference scheme that estimates the solution of space fractional diffusion equation with the Caputo fractional derivative. The proposed scheme's stability, and convergence are proved. To assess the efficiency of this program, a set of tests is carried out. The results of these tests demonstrate the reliability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ -weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O (h 2 + (Δ t) 2) . Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques. [ABSTRACT FROM AUTHOR]

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid's unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]