Search Results (11)

This paper adapts a semi-analytical framework the Laplace-Residual Power Series Method (LRPSM) to solve the time-fractional Allen–Cahn equation under the Caputo derivative. While the classical Allen–Cahn model successfully describes phase separation, its fractional counterpart is essential for capturing sub-diffusive memory effects in complex heterogeneous materials. However, the interplay between the non-local fractional temporal operator and the cubic nonlinearity of the bistable double-well potential creates significant computational bottlenecks for conventional time-domain series solvers. The proposed approach projects the governing fractional partial differential equation into the Laplace domain, systematically replacing the computation of iterative fractional derivatives with the algebraic evaluation of asymptotic limits at infinity. Furthermore, the nonlinear cubic interactions are managed through Laplace-space convolution theorems. The structural convergence of this approach is evaluated against multi-scenario one-dimensional phase transitions. Graphical analyses, featuring 2D profile trajectories and 3D spatiotemporal surface mappings, visually illustrate the retarded interfacial propagation driven by fractional memory. Ultimately, this study presents the LRPSM as an applicable, continuous mathematical tool for approximating anomalous diffusion in the specific phase-field dynamics evaluated herein. Full article

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. Full article

This study introduces a multidimensional mathematical model and a robust numerical algorithm with second-order accuracy for modeling the complex coupled processes of heat and moisture transfer with gas-pressure-driven flow, based on time-fractional differential equations (with Caputo derivatives of order 0 < α ≤ 1), which capture the memory effects and anomalous diffusion inherent in heterogeneous porous media. The proposed model integrates conductive and convective heat transfer; moisture diffusion and phase change; and pressure dynamics within the pore space and their bidirectional couplings. It also incorporates environmental interactions through boundary conditions for heat and moisture exchange with the ambient air; internal heat and moisture release; transient influx of solar radiation; and material heterogeneity, where all transport coefficients are spatially variable functions. To solve this nonlinear and coupled system, we developed a high-order, stable finite-difference scheme. The numerical algorithm employs an alternating direction-implicit approach, which ensures computational efficiency while maintaining numerical stability. We demonstrate the algorithm’s capability through numerical simulations that monitor and predict the spatiotemporal evolution of coupled transport temperature, moisture content, and pressure fields. The results reveal how heterogeneity, diurnal solar radiation, and internal sources create localized hot spots, moisture accumulation zones, and pressure gradients that significantly influence the overall dynamics of storage and drying processes. Full article

The main objective of this study was to identify the undetermined source term (ST) in a fractional space-time scattering equation with multiple terms, using data obtained from the most recent observations. To address this complex problem, we reformulated the equation by adopting a regularization-based optimization approach. This methodology not only makes it possible to determine the existence of a single minimum solution, but also to assess its stability. In the numerical context, we estimate and approach the function (ST) by applying the Levenberg–Marquardt regularization method, a powerful tool for solving inverse problems. In order to demonstrate the effectiveness of the proposed approach, we performed numerical simulations in one-dimensional and two-dimensional scenarios. These simulations illustrate our method’s ability to process complex data and provide accurate and stable solutions. Through this extended approach, we aimed to discover the single source term in a multi-term space-time fractional scattering equation, ensuring robust and reliable results, supported by the most recent observational data. Full article

This paper is focused on the inverse problem of identifying the space-dependent source function and initial value of the time fractional nonhomogeneous diffusion-wave equation from noisy final time measured data in a multi-dimensional case. A mollification regularization method based on a bilateral exponential kernel is presented to solve the ill-posedness of the problem for the first time. Error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical experiments of interest show that our proposed method is effective and robust with respect to the perturbation noise in the data. Full article

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables. Full article

Significant deviations from standard Boltzmann scaling, which corresponds to normal or Fickian diffusion, have been observed in the literature for water movement in porous media. However, as demonstrated by various researchers, the widely used conventional Richards equation cannot mimic anomalous diffusion and ignores the features of natural soils which are heterogeneous. Within this framework, governing equations of transient water flow in porous media in fractional time and multi-dimensional fractional soil space in anisotropic media were recently introduced by the authors by coupling Brooks–Corey constitutive relationships with the fractional continuity and motion equations. In this study, instead of utilizing Brooks–Corey relationships, empirical expressions, obtained by least square fits through hydraulic measurements, were utilized to show the suitability of the proposed fractional approach with other constitutive hydraulic relations in the literature. Next, a finite difference numerical method was proposed to solve the fractional governing equations. The applicability of the proposed fractional governing equations was investigated numerically in comparison to their conventional counterparts. In practice, cumulative infiltration values are observed to deviate from conventional infiltration approximation, or the wetting front through time may not be consistent with the traditional estimates of Richards equation. In such cases, fractional governing equations may be a better alternative for mimicking the physical process as they can capture sub-, super-, and normal-diffusive soil water flow processes during infiltration. Full article

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method. Full article

In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies illustrating a wide spectrum of its applications are presented. The first case study deals with the scale-invariant solutions to a one-dimensional time-fractional diffusion-wave equation that can be represented in terms of the Wright function of the second kind and the four-parameters Wright function of the second kind. In the second case study, we consider a subordination formula for the solutions to a multi-dimensional space-time-fractional diffusion equation with different orders of the fractional derivatives. The kernel of the subordination integral is a special case of the four-parameters Wright function of the second kind. Finally, in the third case study, we shortly present an application of an operational calculus for a composed Erdélyi-Kober fractional operator for solving some initial-value problems for the fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. In particular, we present an example with an explicit solution in terms of the four-parameters Wright function of the second kind. Full article

The fundamental solution to the multi-dimensional space-time fractional diffusion equation is studied by applying the subordination principle, which provides a relation to the classical Gaussian function. Integral representations in terms of Mittag-Leffler functions are derived for the fundamental solution and the subordination kernel. The obtained integral representations are used for numerical evaluation of the fundamental solution for different values of the parameters. Full article

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions. Full article