This study presents a high-order numerical scheme for solving the generalized fractional-order Burgers-Huxley equation, a nonlinear evolutionary PDE combining integer-order spatial derivatives with temporal fractional derivatives. The model captures essential features of reaction-diffusion-convection systems and neural pulse propagation dynamics. We develop a fourth-order compact finite difference method for spatial discretization coupled with a Grünwald-Letnikov approximation for the fractional time derivative. The resulting nonlinear system is solved using an efficient iterative algorithm. Rigorous analysis demonstrates that the proposed method is unconditionally stable and achieves $\mathcal{O}(\tau^3 + h^4)$ convergence, where $\tau$ and $h$ represent temporal and spatial step sizes, respectively. Numerical experiments confirm the theoretical results and illustrate the scheme's accuracy through multiple test cases, demonstrating its effectiveness in handling this class of fractional PDEs.

In this article, we present a highly accurate technique for the numerical solution of the variable-order time-fractional Burgers-Huxley equation. The original equation is first discretized in the temporal and spatial directions. The third-order weighted-shifted Gru & uml;nwald-Letnikov and the fourth-order compact finite difference methods are used. We then formulate a nonlinear system of algebraic equations using the fully discretized version of the problem. The derived nonlinear system is solved by utilizing an iterative algorithm. The analysis of solvability, stability, and convergence of the method is also addressed. The method achieves a convergence rate of four in the spatial direction and three in the temporal direction. Moreover, it is a low-cost computational method and easy to implement. Finally, various illustrative examples are solved to verify the accuracy of the proposed method.

Time-delay fractional optimal control problems (OCPs) are an important research area for developing effective control and optimization strategies to address complex phenomena occurring in various natural sciences, such as physics, chemistry, biology, and engineering. By considering fractional OCPs with time delays, we can design control strategies that take into account the system's history and optimize its behavior over a given time horizon. However, applying the Pontryagin principle of maximization to solve these problems leads to a boundary value problem (BVP) that includes delay and advance terms, making analytical solutions difficult and demanding. To address this issue, this paper presents a precise finite difference formula to solve the aforementioned advance-delay BVP numerically. The suggested approximate method's error analysis and convergence properties are provided, and several illustrative examples demonstrate the applicability, validity, and accuracy of the proposed approach. Simulation results confirm the proposed technique's advantages for the optimal control of delay fractional dynamical equations.

In this paper, we propose a fourth-order compact discretization method for solving a secondorder boundary value problem governed by the nonlinear Fredholm integro-differential equations. Using an efficient approximate polynomial, a compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some numerical examples are presented to demonstrate its approximation accuracy and computational efficiency, as well as to compare the derived results with those obtained in the literature.

Abstract The aim of this paper is to investigate some V0-stable numerical methods based on the θ-method for solving the Volterra integral equations (VIEs) of the second kind. The proposed V0-stable methods have the second- and third-order of acuuracy, and guarantee the V0-stability property for VIEs. Numerical simulations show that these methods are succseeful for various types of the VIEs and their computational cost are less than the other methods in literature.

Abstract The aim of this paper is to investigate some second- and third-order numerical methods for solving the Volterra integral equations (VIEs) of the second kind. The proposed numerical methods preserve the V_0-stability property for both single and system of the VIEs. Numerical simulations show that these methods are successful for various types of the VIEs and their computational cost are less than the other methods in literature.

This paper is concerned with a highly accurate numerical scheme for a class of one and two-dimensional time-fractional advection-reaction-subdiffusion equations of variable order alpha(x, t) is an element of (0, 1). For the spatial and temporal discretization of the equation, a fourth order compact finite difference operator and a third-order weighted-shifted Grunwald formula are applied, respectively. The stability and convergence of the present scheme are addressed. Some extensive numerical experiments are performed to confirm the theoretical analysis and high-accuracy of this novel scheme. Comparisons are also made with the available schemes in the literature. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

The main focus of this paper is on studying an order optimal regularization scheme based on the Meyer wavelets method to solve the analytic continuation problem in the high dimensional complex domain Omega := {x + iy is an element of C-N : x is an element of R-N, parallel to y parallel to <= parallel to y(0)parallel to, y, y(0) is an element of R-+(N)}. This problem is exponentially ill-posed and suffers from the Hadamard's instability. Theoretically, we first provide an optimal conditional stability estimate for the proposed original problem. Applying the Meyer wavelets, an order optimal regularization scheme is then developed to stabilize the considered ill-posed problem. Some sharp error estimates of the Holder-Logarithmic type controlled by the Sobolev scale under an a priori information are also derived. The provided error estimates are of the order optimal in the sense of Tautenhahn. Finally, some different oneand two-dimensional examples are presented to confirm the efficiency and applicability of the proposed regularization scheme. The comparison results also show that the proposed method is more accurate than the other existing methods in the literature. (c) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

In this paper, an accurate numerical method based on the cosine-trigonometric basis functions is developed to solve the Fredholm integral equations of the second kind. By using the proposed method, the presented equation is converted into a system of algebraic equations. The convergence analysis of the proposed method is also investigated. To demonstrate the efficiency of the proposed method, the numerical simulations of various types of one- and two-dimensional examples are prepared. Comparative results show that this method is accurate than the other existing methods in the literature. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

The aim of this paper is to investigate an efficient numerical method based on a novel shifted piecewise cosine basis for solving Volterra-Fredholm integral equations of the second kind. Using operational matrices of integration for the proposed basis functions, this integral equation is transformed into a system of nonlinear algebraic equations. The convergence and error analysis of the proposed method are studied. Some comparative results are provided to verify the efficiency of the presented method. (C) 2019 Elsevier Inc. All rights reserved.

This manuscript deals with a regularization technique for a generalized space-fractional backward heat conduction problem (BHCP) which is well-known to be extremely ill-posed. The presented technique is developed based on the Meyer wavelets in retrieving the solution of the presented space-fractional BHCP. Some sharp optimal estimates of the Hölder-Logarithmic type are theoretically derived by imposing an a-priori bound assumption via the Sobolev scale. The existence, uniqueness and stability of the considered problem are rigorously investigated. The asymptotic error estimates for both linear and non-linear problems are all the same. Finally, the performance of the proposed technique is demonstrated through one- and two-dimensional prototype examples that validate our theoretical analysis. Furthermore, comparative results verify that the proposed method is more effective than the other existing methods in the literature.

The aim of this paper is to develop an accurate discretization technique to solve a class of variable-order fractional (VOF) reaction-diffusion problems. In the spatial direction, the problem is first discretized by using a compact finite difference operator. Then, a weighted-shifted Grunwald formula is applied for the temporal discretization of fractional derivatives. To solve the derived nonlinear discrete system, an accurate iterative algorithm is also formulated. The solvability, stability and L-2-convergence of the proposed scheme are derived for all variable-order alpha(t) is an element of (0, 1). The proposed method is of accuracy-order O(tau(3) + h(4)), where tau and h are temporal and spatial step sizes, respectively. Through some numerical simulations, the theoretical analysis and high-accuracy of the proposed method are verified. Comparative results also indicate that the accuracy of the new discretization technique is superior to the other methods available in the literature. Finally, the feasibility of the proposed VOF model is demonstrated by using the reported experimental data. (C) 2018 Elsevier B.V. All rights reserved.

SummaryThe principle result of this paper is the following disturbance rejection control scheme for a class of nonlinear dynamical systems. By using the internal model principle, the problem of disturbance damping control is converted into a nonlinear quadratic regulator (NQR) problem for an undisturbed augmented system. Then, an iterative technique is designed to solve this NQR problem effectively. The proposed iterative method is also extended through the use of a nonlinear model predictive control in an offline framework. In this case and in the presence of unknown disturbances, the Lyapunov stability of the closed‐loop system is guaranteed. Numerical simulations and comparative results verify the effectiveness of the proposed approach.

The aim of this study is to develop an efficient iterative approach for solving a class of time-delay optimal control problems with time-varying delay and external persistent disturbances. By using the internal model principle, the original time-delay model with disturbance is first converted into an augmented system without any disturbance. Then, we select a quadratic performance index for the augmented system to form an undisturbed time-delay optimal control problem. The necessary optimality conditions are then derived in terms of a two-point boundary value problem involving advance and delay arguments. Finally, a fast iterative algorithm is designed for the latter advance-delay boundary value problem. The convergence of the new iterative technique is also investigated. Numerical simulations verify that the proposed approach is efficient and provides satisfactory results.

This paper presents a class of semi-implicit finite difference weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear heat equation. For the discretization of second-order spatial derivatives, a sixth-order modified WENO scheme is directly implemented. This scheme preserves the positivity principle and rejects spurious oscillations close to non-smooth points. In order to admit large time steps, a class of implicit Runge–Kutta methods is used for the temporal discretization. The implicit parts of these methods are linearized in time by using the local Taylor expansion of the flux. The stability analysis of the semi-implicit WENO scheme with 3-stages form is provided. Finally, some comparative results for one-, two- and three-dimensional PDEs are included to illustrate the effectiveness of the proposed approach.

Using the Pontryagin's maximum principle for a time-delayed optimal control problem results in a system of coupled two-point boundary-value problems (BVPs) involving both time-advance and time-delay arguments. The analytical solution of this advance-delay two-point BVP is extremely difficult, if not impossible. This paper provides a discrete general form of the numerical solution for the derived advance-delay system by applying a finite difference.-method. This method is also implemented for the infinite-time horizon time-delayed optimal control problems by using a piecewise version of the.-method. A matrix formulation and the error analysis of the suggested technique are provided. The new scheme is accurate, fast and very effective for the optimal control of linear and nonlinear time-delay systems. Various types of finite-and infinite-time horizon problems are included to demonstrate the accuracy, validity and applicability of the new technique. (C) 2017 Elsevier B. V. All rights reserved.

This manuscript mainly focuses on the construction, dynamic analysis and control of a new fractional-order financial system. The basic dynamical behaviors of the proposed system are studied such as the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits of state variables and the intervals of system parameters. It is shown that the system exhibits hyperchaotic behavior for a number of system parameters and fractional-order values. To stabilize the proposed hyperchaotic fractional system with uncertain dynamics and disturbances, an efficient adaptive sliding mode controller technique is developed. Using the proposed technique, two hyperchaotic fractional-order financial systems are also synchronized. Numerical simulations are presented to verify the successful performance of the designed controllers.

The aim of this manuscript is to investigate an accurate discretization method to solve the one-, two-, and three-dimensional highly nonlinear Bratu-type problems. By discretization of the nonlinear equation via a fourth-order nonstandard compact finite difference formula, the considered problem is reduced to the solution of a highly nonlinear algebraic system. To solve the derived nonlinear system, a modified nonlinear solver is used. The new scheme is accurate, fast, straightforward and very effective to find the lower and upper branches of the Bratu’s problem. Numerical simulations and comparative results for the one-, two-, and three-dimensional cases verify that the new technique is easy to implement and more accurate than the other existing methods in the literature.

The aim of this manuscript is to investigate an efficient iterative approach for the nonlinear fractional optimal control problems affected by the external persistent disturbances. For this purpose, first the internal model principle is employed to transform the fractional dynamic system with disturbance into an undisturbed system with both integer- and fractional-order derivatives. The necessary optimality conditions are then reduced into a sequence of linear algebraic equations by using a series expansion approach and the Grunwald-Letnikov approximation for the fractional derivatives. The convergence of the latter sequence to the optimal solution is also studied. In addition, an iterative algorithm designing the suboptimal control law is presented. Numerical simulations confirm that the new approach is efficient to reject the external disturbance and provides satisfactory results compared to the other existing methods. (C) 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.