Search Results (18)

This work presents an efficient analytical framework based on the Natural Residual Power Series Method (NRPSM) for solving several forms of the time-fractional Kuramoto–Sivashinsky equation with the Caputo derivative. The proposed method avoids discretization and linearization while producing rapidly convergent analytical series solutions. Earlier residual power series treatments assert convergence under a contractivity assumption without verifying it for the equation at hand. We close this gap by deriving an explicit formula for the contraction constant directly from the problem data, so the convergence criterion is checkable before any computation begins. A rigorous theoretical analysis is established through explicit contraction conditions, convergence proofs in the Sobolev space $H^4\left(\mathbb{R}\right)$, and an explicit geometric-type error estimate that quantifies how the fractional order governs the convergence rate through two competing effects, without presuming a uniform direction of influence. Stability with respect to perturbations in the initial data is also proven using a fractional Gronwall inequality. Numerical results demonstrate excellent agreement with exact and previously published solutions, achieving very small absolute errors using only a few series terms. The obtained results confirm that the NRPSM is an accurate, stable, and computationally efficient approach for nonlinear fractional evolution equations. Full article

This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions. Full article

This paper studies nonlinear time-fractional reaction–diffusion equations with Caputo memory and non-local integral boundary conditions on a bounded interval. The aim is to formulate a boundary-compatible well-posedness framework and to construct a high-order temporal approximation that can be coupled with a constraint-preserving spectral spatial discretization. The analytical part proves boundedness of the non-local boundary functionals, states compatibility assumptions, and introduces a finite-dimensional nondegeneracy condition for an explicit polynomial lifting. Under a sectorial non-local elliptic realization and a global Lipschitz reaction term, existence, uniqueness, stability, and continuous dependence of mild solutions are obtained by fractional resolvent estimates and fractional Gronwall inequalities. The main novelty is the combined construction of an explicit polynomial lifting for integral boundary constraints, a constraint-preserving Legendre–Galerkin basis, and a high-order Beta-window temporal quadrature together with a discrete stability condition that accounts for sign-changing weights. The numerical evidence shows high-order behavior for smooth Caputo benchmarks, accurate enforcement of the non-local boundary constraints, and improved accuracy over the classical $L1$ approximation in the reported tests. The stability discussion identifies the discrete coercivity condition required for the sign-changing Beta-window weights. Full article

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (${\beta }_{\mathrm{NM}}=1/4$) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders $\alpha \ne \beta$, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory. Full article

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable $v=-{\epsilon }^2\Delta u+f(u)$, the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only $1/60$ of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods. Full article

In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable $\sigma =\Delta u$, the original fourth-order model is reformulated into a lower-order coupled system. The first-order time derivative and the time fractional derivative are discretized by using the BDF2 formula and the weighted and shifted Grünwald difference (WSGD) formula, respectively. Then, a fully discrete MFVE scheme is constructed by using the primal and dual grids. The existence and uniqueness of a solution for the MFVE scheme are proven based on the matrix theories. The scheme’s unconditional stability is rigorously derived by using the Gronwall inequality in detail. Moreover, the optimal error estimates for u in the discrete $L^{\infty }\left(L^2(\Omega )\right)$ and $L^2\left(H^1(\Omega )\right)$ norms and for $\sigma$ in the discrete $L^2\left(L^2(\Omega )\right)$ norm are obtained. Finally, three numerical examples are given to confirm its feasibility and effectiveness. Full article

This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the linear case, we provide an explicit solution formula involving discrete Mittag-Leffler functions and establish its stability properties. In the nonlinear case, we concentrate on delayed neutral Hilfer fractional difference equations, a class of systems that appears to be unexplored in the existing literature with respect to Ulam–Hyers stability. In particular, for the linear case, the absolute difference between the solution of the linear Hilfer fractional difference equation and the solution of the corresponding perturbed equation is bounded by the function of $\epsilon$ when the perturbed term is bounded by $\epsilon$. In the case of the neutral fractional delayed Hilfer difference equation, the absolute difference is bounded by a constant multiple of $\epsilon$. Our results fill this gap by offering novel stability criteria. We support our theoretical findings with illustrative numerical examples and simulations, which visually confirm the predicted stability behavior and demonstrate the applicability of the results in discrete fractional dynamic systems. Full article

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an exponential Euler difference approach to examine the properties of nonlocal discrete-time oscillators with Mittag–Leffler kernels and piecewise features, with the aim of providing insights into a continuous-time nonlocal nonlinear system. By employing impulsive equations of variations in constants with different forms in conjunction with the Gronwall inequality, a controller that is capable of instantaneously responding and stabilizing the nonlocal discrete-time oscillator is devised. This controller is realized through an associated algorithm. As a case study, the primary outcome is applied to a problem of impulsive stabilization in nonlocal discrete-time Chua’s oscillator. This article presents a stabilizing algorithm for piecewise nonlocal discrete-time oscillators developed using a novel impulsive approach. Full article

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results. Full article

In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral method using the Laguerre function in space. We obtained an $\alpha$-robust discrete Gronwall inequality and the a priori error estimation of the numerical solution. Then, the existence and uniqueness of the numerical solution are discussed. Next, we present the $\alpha$-robust stability and convergence of the fully discrete scheme, where the convergence was obtained based on the regularity conditions of the exact solution. A numerical example demonstrates the validity of the theoretical results. Full article

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied. Full article

In this paper, we address the finite-time stability problem of Caputo nabla fractional-order switched linear systems with $\alpha \in (0,1)$. Firstly, the monotonicity of the discrete Mittag-Leffler function is proposed. Secondly, under the per-designed switching rules, the form of the solution for Caputo nabla fractional-order switched linear systems is obtained by using the discrete unit step function. On the above basis, some sufficient conditions of finite-time stability for Caputo nabla fractional-order switched linear systems are proposed, according to the discrete Grönwall inequality and the monotonicity of the discrete Mittag-Leffler function. Finally, simulation verification is carried out via three numerical examples. Full article

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order $\alpha \in (0,1)$. Considering the weak singularity of the solution $u(\mathbf{x},t)$ at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as $t\to 0^{+}$ albeit the function itself being right continuous at $t=0$, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-$1_{\sigma }$ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results. Full article

In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time $t=0$. The Caputo time-fractional derivative with derivative order $\alpha \in (0,1)$ is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., $(2-\alpha )$-order accurate in time and $(k+1)$-order accurate in space, when piece-wise polynomials of degree at most k are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the $L2$-$1_{\sigma }$ formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results. Full article

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from $\mathcal{O}\left(N_s^2\right)$ to $\mathcal{O}\left(N_s\right)$ and the computational complexity from $\mathcal{O}\left(N_s^3\right)$ to $\mathcal{O}(N_{s}log{N}_s)$ in each iterative step, where $N_s$ is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method. Full article