This study is devoted to the development and analysis of a high-order compact finite difference method for nonlinear partial integro-differential equations of fractional order. The Caputo-type fractional derivative in time is approximated using the L1 scheme on graded temporal meshes, which effectively handles the weak singularity at t = 0, while the Riemann-Liouville integral term is evaluated through a quadrature formula that preserves the scheme's high-order accuracy. In space, a fourth-order compact finite difference scheme is employed, yielding substantially higher accuracy than the standard second-order methods. A matrix formulation of the fully discrete system is developed, and rigorous stability and convergence results are established. Theoretical results indicate that the method achieves a temporal convergence rate of (9(min{gyp, 2-S}), where gyis the grading parameter of the temporal mesh, p is the regularity parameter, and S represents the order of the fractional derivative, together with fourth-order spatial accuracy. The main contribution is the integration of graded temporal discretization for the fractional derivative, accurate quadrature for the fractional integral, and a fourth-order compact spatial scheme, leading to a stable and efficient high-order algorithm for proposed fractional models. Moreover, unlike most existing studies that are restricted to two-dimensional settings, our approach naturally extends to three-dimensional problems. Numerical experiments for several nonlinear fractional equations confirm the theoretical results and demonstrate the robustness, accuracy, and efficiency of the present scheme.

Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally in fractional models, weakly singular integral equations, and semi-infinite-domain approximations with limited regularity near the origin. The main contribution of this work is the construction and analysis of a fractional Laguerre approximation framework tailored to nonsmooth functions on the half-line. We establish projection and interpolation error estimates in nonuniformly weighted Sobolev space. These estimates clarify how the fractional parameter adapts the approximation space to the regularity of nonsmooth functions and improves the resulting convergence behavior. We further introduce a generalized fractional Laguerre family with an additional algebraic parameter, which gives greater flexibility in controlling both the approximation space and the underlying weight. Numerical experiments confirm the theoretical estimates and demonstrate the advantage of the proposed functions over standard Laguerre-type approximations.

We provide a unified analysis framework for discretized Volterra integrodifferential equations by considering the 9-type convolution quadrature, where different 9 corresponds to different schemes. We first derive the long-time lCO stability of discrete solutions, and then prove a discrete Wiener-Levy theorem to support the analysis of long-time l1 stability. The methods we adopt include the integral transforms in the Stieltjes sense, the complex analysis techniques, and a linear algebra approach for an indirect estimate of intricate terms. Meanwhile, we relax the commonly-used regularity assumption of the initial data in the literature by novel treatments. Numerical simulations are performed to substantiate the theoretical findings.

. This study aims to determine the source term within a subdiffusion model using an artificial neural network approach, leveraging additional data. The core strategy involves replacing the unknown source term with a neural network, thereby converting the inverse source problem into an optimal control problem. By employing this approach, we establish the existence of an optimal solution for the control problem and compute the associated optimality conditions using the alternating direction multiplier method. Numerical solutions are derived via this proposed methodology, showcasing its effectiveness through a series of numerical tests on both regular and singular examples. Our results demonstrate the efficacy of the artificial neural network method, particularly evident in its comparison with established techniques like gradient descent and alternating direction multiplier method. This comparative analysis reinforces the strength and reliability of the artificial neural network-based approach in solving the source term determination problem within the sub diffusion model.

Tempered fractional derivatives generalize classical fractional operators by incorporating an exponential tempering factor, which enables more realistic modeling of nonlocal processes with finite memory. In particular, left-sided tempered fractional derivatives capture dynamics influenced by past states and are therefore well suited for initial value problems. However, achieving high accuracy is challenging because solutions of such problems typically exhibit mild singular behavior near the initial point, which significantly degrades the convergence of standard spectral schemes. To address this difficulty, we develop a new initial-point adaptive spectral method that enriches and refines the approximation space in a neighborhood of the initial singularity. The proposed approach attains high accuracy for both rational and irrational tempered fractional orders. We also provide a rigorous L^2_ϰ ^γ -1,0,ν -error convergence analysis to establish the efficiency, and robustness of the method. Numerical experiments confirm the theoretical findings and demonstrate that the proposed framework offers a reliable tool for a broad class of left-sided tempered fractional differential equations.

We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally reflects the terminal-endpoint singular behaviour produced by weakly singular kernels. We develop the basic approximation theory for the proposed backward orthogonal basis, including weighted projection estimates, Gauss-type interpolation estimates, inverse inequalities, and stability bounds for the associated weakly singular adjoint integral operator. The error analysis and numerical results show that the proposed backward Jacobi method is particularly suitable for solutions with terminal-endpoint weak singularities and can recover high-order convergence rates that are typically lost when usual polynomial approximations are applied directly to such weakly regular solutions.

We develop a systematic framework for incorporating perturbative correction terms into classical finite difference schemes for Allen-Cahn type stochastic partial differential equations. Three distinct correction approaches are introduced, conceptually motivated by perturbative quantum field theory, quantum coherent state evolution, and WKB (Wentzel-Kramers-Brillouin) barrier penetration theory. These quantum-inspired perturbative method (QIPM) corrections function as classical perturbations executing entirely on conventional hardware; quantum-mechanical formalism serves only as a design principle for constructing specific functional forms of correction terms. The primary novelty of this work lies in (i) a generic convergence-preservation theorem establishing sufficient conditions on correction magnitude for any perturbative correction to maintain the base scheme's accuracy order, and (ii) a systematic translation methodology from quantum-mechanical analogies to explicit, implementable finite difference corrections with rigorous parameter bounds. Through convergence analysis, we demonstrate that appropriately parametrized corrections preserve the accuracy of the underlying numerical scheme, provided the solution possesses sufficient regularity and the parabolic scaling constraint Delta t=O(h2) holds. Numerical experiments on a spatially discretized domain over a finite time horizon using spatially correlated noise reveal that the anharmonic oscillator correction achieves exceptional accuracy with modest computational overhead, while the amplitude encoding correction provides intermediate accuracy with negligible timing cost. The tunneling-inspired correction exhibits higher error for smooth initial conditions, indicating strong problem-dependence. While these methods enhance accuracy in specific scenarios, genuine speedups on classical hardware are not achieved. The primary value lies in establishing systematic methodologies for perturbative correction design and developing theoretical foundations for future hybrid computational approaches.

This study investigates a time-fractional optimal control problem for the Bergman minimal model of type 1 diabetes by employing the Caputo derivative of order q is an element of (0,1) to incorporate memory effects into the glucose-insulin dynamics. First, the well-posedness of the fractional-order model is established by proving the existence, uniqueness, boundedness, and positivity of its solutions. Subsequently, an optimal control problem describing a normalized therapeutic intervention is formulated, and the existence of an optimal control is rigorously proved. By applying the fractional Pontryagin maximum principle, the corresponding optimality system is derived, and the optimal control law is explicitly characterized. The obtained results demonstrate that the proposed control strategy effectively stabilizes the glucose dynamics around the desired equilibrium while reducing excessive insulin administration. In comparison with the classical integer-order model, the fractional formulation provides a more realistic representation of delayed physiological responses. Finally, numerical simulations, carried out using a predictor-corrector scheme together with a forward-backward sweep algorithm, confirm the effectiveness and reliability of the proposed approach for blood glucose regulation.

This paper presents a novel and efficient numerical methodology for solving time-fractional partial integro-differential equations with time delay, employing a combination of finite difference, spectral, and finite block methods on both square and non-rectangular regions. The Caputo derivative is discretized employing the L2-1_σ scheme, while a L–type approximation is applied to the Riemann–Liouville fractional integral at the n+σ time level. For square domains, a fully–discrete scheme is constructed using Chebyshev spectral methods via the differentiation matrix. For irregular geometries, the finite block method is developed to handle domain complexity effectively. Theoretical analyses of stability and convergence are investigated, and a series of numerical experiments demonstrate the accuracy and computational efficiency of the proposed method.

In this work, we construct and analyze a novel endpoint-adaptive fractional spectral collocation method for nonlinear terminal value problems involving tempered Caputo derivatives. Unlike many existing approaches that rely on fractional transformations or variable changes, which are typically restricted to rational fractional orders and often lose accuracy near singular endpoints, our framework directly employs a new class of fractional basis functions. This construction provides flexibility in treating both rational and irrational fractional orders without the need for auxiliary smoothing transformations, while naturally capturing the endpoint singularities that arise in terminal value formulations. Moreover, the method provides a unified framework for handling non-smooth solutions, a setting in which conventional polynomial-based schemes generally fail to maintain spectral accuracy. Rigorous error analysis establishes convergence, and numerical experiments demonstrate that the proposed method delivers superior accuracy and efficiency. Overall, the endpoint-adaptive method fills an important gap in the literature by providing a robust and versatile computational framework for challenging tempered fractional models with irrational orders and non-smooth behavior.

Distributed-order fractional differential equations are defined by integrating over a continuum of fractional differentiation orders. This structure can represent complex memory effects and dynamics that evolve across multiple time scales. Yet the distributed-order operator is expensive for classical numerical schemes. It also does not fit naturally within automatic differentiation workflows used in neural network solvers. This paper presents an efficient tensor-based formulation that approximates the distributed-order integral in a direct and unified way. We first build an operational matrix for the Caputo fractional derivative and evaluate the distributed-order integral using Gauss-Legendre quadrature in the fractional-order variable. We then promote the resulting operator to a tensor-product structure that is well suited to parallel computation, allowing evaluation of the full distributed-order operator in a single step. We embed the tensorized operator into a physics-informed Kolmogorov-Arnold network that uses fractional B-spline basis functions. The result is a meshfree solver for distributed-order fractional differential equations. Numerical experiments confirm accuracy and robustness on benchmark problems that include inverse problems, high-dimensional systems on irregular domains, and solutions with limited smoothness.

In this paper, we construct and analyze a linearized L1-Galerkin Legendre spectral method for solving two-dimensional time-fractional diffusion equations with delay. The approach combines the L1 temporal discretization of the Caputo derivative with a Legendre spectral approximation in space, while nonlinear source terms are efficiently handled through a linearization strategy. To further enhance computational performance, we employ matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Rigorous stability and convergence analyses are carried out using discrete fractional Gr & ouml;nwall and fractional Halanay inequalities, establishing unconditional stability and spectral error estimates. Numerical experiments confirm the theoretical predictions, demonstrating spectral accuracy in space and (2-#)-order accuracy in time, as well as validating the robustness and efficiency of the proposed method across different fractional orders and delay parameters.

We present a novel Fractional Power Iterative regularization method for solving Caputo-Hadamard fractional inverse source problems arising in anomalous diffusion applications. The proposed method addresses the inherent ill-posedness of reconstructing unknown source terms from noisy boundary observations by combining fractional power regularization operators with iterative refinement strategies. Rigorous convergence analysis establishes superlinear convergence rates and optimal error bounds under appropriate smoothness assumptions, with spectral analysis revealing exponential decay characteristics that significantly outperform classical Tikhonov regularization. Comprehensive numerical experiments demonstrate the method’s superiority across multiple performance metrics, showing substantial improvements in reconstruction accuracy and optimal linear scaling behavior with noise levels, while maintaining exceptional spectral preservation capabilities even under challenging noise conditions. Environmental applications to groundwater contamination transport modeling demonstrate the practical significance of fractional diffusion frameworks, where the Caputo-Hadamard operator captures memory effects critical for accurate prediction of contaminant plume evolution in heterogeneous aquifers that classical models significantly underestimate in terms of cleanup timeframes and barrier performance requirements. The research establishes this approach as the current state-of-the-art for fractional inverse source problems, providing essential tools for environmental engineering applications including remediation design, exposure assessment, and long-term monitoring strategies in complex groundwater systems.

This research presents an innovative stochastic modeling approach for simulating pollutant dispersion through fractional stochastic drift-flux (FSDF) models, combining fractional calculus with probabilistic processes to address anomalous diffusion and environmental uncertainties in heterogeneous systems. The framework employs the Karhunen-Loève expansion with a mixed noise methodology, demonstrating superior performance compared to conventional white noise simulations by achieving enhanced realism scores and capturing substantial variance with minimal computational modes for improved efficiency. Temporal fractional derivatives are discretized using the L1-algorithm, while spatial advection and diffusion components are addressed through upwind and central difference schemes, respectively. Comprehensive stability analysis, validated through energy estimation techniques, establishes the framework’s reliability across different fractional orders, with reduced fractional values promoting stability via enhanced memory effects. Computational experiments confirm the model’s precision in representing continuous source emission and natural decay processes, effectively capturing realistic concentration behavior under stochastic conditions. The FSDF model methodology provides substantial improvements for environmental risk assessment, including urban air quality evaluation, industrial emission monitoring, and stochastic environmental analysis, delivering a robust probabilistic framework for uncertainty quantification and environmental decision-making. Planned developments including multi-dimensional domain applications and real-time data incorporation are expected to broaden its utility in sophisticated environmental modeling applications.

This work presents a new and efficient numerical framework for solving three-dimensional time-fractional diffusion and mobile–immobile equations in the Caputo sense. The method is formulated using four-variable Jacobi polynomials, constructed systematically via the Kronecker product of one-dimensional Jacobi bases to accurately represent the multidimensional nature of the governing equations. Within a pseudo-operational collocation formulation, these polynomials enable a highly accurate and computationally efficient approximation of the fractional operators in both temporal and spatial directions. From the theoretical standpoint, the existence and uniqueness of the approximate solution are rigorously established through Schauder’s fixed-point theorem. Furthermore, the Ulam–Hyers stability of the numerical solution is verified, demonstrating the robustness of the method with respect to perturbations in the input data. To reinforce the reliability of the approach, an explicit error bound for the residual function is derived in a Jacobi-weighted Sobolev space, offering a firm analytical basis for assessing convergence. Numerical experiments confirm that the proposed approach achieves superior accuracy and efficiency, highlighting its potential as a powerful tool for high-dimensional fractional partial differential equations.

In this work, we consider a numerical approximation for the two-dimensional fractal mobile/immobile transport equation with weakly singular solutions, where the time first-order derivative is discretized by the backward Euler method, and the Caputo fractional derivative is approximated by the L1 scheme on a uniform mesh. The fully discrete ADI scheme is established by adding a high-order term. The stability and the convergence analyses of the fully discrete ADI scheme are analyzed in L2-norm and H1-norm. The numerical results show that the error estimates are sharp.

In the present work, we investigate a distributed-order time–fractional partial integro–differential equation (DOTFPIDE) in one and higher dimensions, employing the graded temporal mesh for discretization. The distributed–order time–fractional derivative is approximated using the nonuniform L_1 formula, while the fractional integral term is discretized via a product integration approach. For spatial discretization, Chebyshev nodes are utilized as collocation points, and a hybrid methodology combining the alternating direction implicit scheme with collocation techniques is applied to address multi–dimensional problems. In irregular computational domains, the finite block method is adopted for two–dimensional DOTFPIDEs. The stability and convergence of the numerical schemes are rigorously analyzed, and computational experiments are conducted to validate their theoretical accuracy and efficiency. These results demonstrate the robustness of the proposed framework in solving complex distributed-order fractional integro–differential equations across diverse geometries.

Fuzzy Volterra integral equations with weakly singular kernels have solutions with singular behavior at the origin. Utilizing spectral methods on such problems with standard (integer-order) basis functions leads to generating approximate solutions with low-order accuracy. This paper deals with improving the accuracy of the spectral Galerkin method which can be applied to such problems by using fractional-order basis functions. New matrix formulation of the proposed method transforms the problem under consideration into a system of algebraic equations with a simple structure. Numerical implementation of the constructed method shows its effectiveness compared to other methods. The convergence analysis of the method is theoretically investigated in a weighted L2-norm.

A determination of the nonsmooth potential parameter in a nonlinear subdiffusion model using terminal observation through a nonsmooth optimal control approach is proposed. This is achieved by converting the inverse problem to an optimal control problem with the fidelity and regularization terms in the cost functional represented by the L1 and TV norms, respectively, in order to account for the nonsmoothness of the potential. The existence of a minimizer for the optimal control problem is established and numerical solutions are obtained using the primal- dual method. The effectiveness of the proposed method is demonstrated via numerical results in comparison with the gradient descent method.