Search Results (475)

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features. Full article

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is developed for a broad class of time-fractional system equations. In particular, we check the validity and effectiveness of the approach by providing three illustrative examples, confirming its accuracy and applicability in solving both linear and nonlinear fractional problems. Full article

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article

We consider an initial value problem for a system of linear fractional differential equations of Caputo type. Using an integral equation reformulation of the underlying problem, we first study the existence, uniqueness and smoothness of its exact solution. Based on the obtained results, a collocation-type method using the central part interpolation approach on the uniform grid is constructed and analyzed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments. Full article

In this manuscript, we study a class of nabla fractional difference equations with summation boundary conditions that depend on a parameter. We construct the Green’s function related to the linear problem and we deduce some of its properties. First, we obtain an upper bound of the sum of it, and use this property to give an existence result for the considered problem based on the Leray–Shauder nonlinear alternative. Then, we establish some bounds on the parameter in which the Green’s function is positive, and by using Krasnoselski–Zabreiko fixed-point theorem, we deduce another existence result. Finally, we give some particular examples in order to demonstrate our primary findings. Full article

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem ${(a+b||(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\left|\right|}_2^2{)(-\Delta )}^{s}u+\lambda u=g\left(u\right)+{\left|u\right|}^{2_s^{*}-2}u$ in ${\mathbb{R}}^3$ with prescribed mass ${\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx=\rho >0,$ where $s\in ({\displaystyle \frac{3}{4}},1),\lambda \in \mathbb{R},2_s^{*}={\displaystyle \frac{6}{3-2s}}$. Under some general growth assumptions imposed on g, we employ minimization of the energy functional on the linear combination of Nehari and Poho$\breve{z}$aev constraints intersected with the closed ball in the $L^2\left({\mathbb{R}}^3\right)$ of radius $\rho$ to prove the existence of normalized ground state solutions to the equation. Moreover, we provide a detailed description for the asymptotic behavior of the ground state energy map. Full article

To alleviate the impact of economic and environmental detriments caused by the increased demands of electric vehicle battery production and disposal, the use of spent batteries in second-life stationary applications such as energy storage for renewable sources or backup power systems, offers many benefits. This paper focuses on reducing the energy consumption cost and greenhouse gas emissions of Internet-of-Things-enabled campus microgrids by installing solar photovoltaic panels on rooftops alongside energy storage systems that leverage second-life batteries, a gas-fired campus power plant, and a wind turbine while considering the potential loads of a prosumer microgrid. A linear optimization problem is derived from the system by scheduling energy exchanges with the Ontario grid through net metering and solved by using Python 3.11. The aim of this work is to support Sustainable Development Goals, namely 7 (Affordable and Clean Energy), 11 (Sustainable Cities and Communities), 12 (Responsible Consumption and Production), and 13 (Climate Action). A comparison between a base case scenario and the results achieved with the proposed scenarios shows a significant reduction in electricity cost and greenhouse gas emissions and an increase in self-consumption rate and renewable fraction. This research work provides valuable insights and guidelines to policymakers. Full article

This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ($LF{L}_{\zeta }^2$) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the $LF{L}_{\zeta }^2$ transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries. Full article

Grey models have attracted considerable attention as a time series forecasting tool in recent years. Nevertheless, the linear characteristics of the differential equations on which traditional grey models rely frequently result in inadequate predictive accuracy and applicability when addressing intricate nonlinear systems. This study introduces a conformable fractional order unbiased kernel-regularized nonhomogeneous grey model (CFUKRNGM) based on statistical learning theory to address these limitations. The proposed model initially uses a conformable fractional-order accumulation operator to derive distribution information from historical data. A novel regularization problem is then formulated, thereby eliminating the bias term from the kernel-regularized nonhomogeneous grey model (KRNGM). The parameter estimation of the CFUKRNGM model requires solving a linear equation with a lower order than the KRNGM model, and is automatically calibrated through the Bayesian optimization algorithm. Experimental results show that the CFUKRNGM model achieves superior prediction accuracy and greater generalization performance compared to both the KRNGM and traditional grey models. Full article

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging. Full article

This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using the homotopy perturbation method (HPM), we construct a homotopy with an embedding parameter to solve this recurrence relation. Our proposed technique is known as the Sumudu homotopy transform method (SHTM), which delivers results after fewer iterations and achieves precise outcomes with minimal computational effort. The proposed technique effectively eliminates the necessity for complex discretization or linearization, making it highly suitable for nonlinear problems. We showcase two numerical cases, along with two- and three-dimensional visualizations, to validate the accuracy and effectiveness of this technique. It also produces rapidly converging series solutions that closely align with the precise results. Full article

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors. Full article

Ignition delay time (IDT) is a critical parameter for evaluating the autoignition characteristics of aviation fuels. However, its accurate prediction remains challenging due to the complex coupling of temperature, pressure, and compositional factors, resulting in a high-dimensional and nonlinear problem. To address this challenge for the complex aviation kerosene RP-3, this study proposes a multi-stage hybrid optimization framework based on a five-input, one-output BP neural network. The framework—referred to as CGD-ABC-BP—integrates randomized initialization, conjugate gradient descent (CGD), the artificial bee colony (ABC) algorithm, and L2 regularization to enhance convergence stability and model robustness. The dataset includes 700 experimental and simulated samples, covering a wide range of thermodynamic conditions: 624–1700 K, 0.5–20 bar, and equivalence ratios φ = 0.5 − 2.0. To improve training efficiency, the temperature feature was linearized using a 1000/T transformation. Based on 30 independent resampling trials, the CGD-ABC-BP model with a three-hidden-layer structure of [21 17 19] achieved strong performance on internal test data: R² = 0.994 ± 0.001, MAE = 0.04 ± 0.015, MAPE = 1.4 ± 0.05%, and RMSE = 0.07 ± 0.01. These results consistently outperformed the baseline model that lacked ABC optimization. On an entirely independent external test set comprising 70 low-pressure shock tube samples, the model still exhibited strong generalization capability, achieving R² = 0.976 and MAPE = 2.18%, thereby confirming its robustness across datasets with different sources. Furthermore, permutation importance and local gradient sensitivity analysis reveal that the model can reliably identify and rank key controlling factors—such as temperature, diluent fraction, and oxidizer mole fraction—across low-temperature, NTC, and high-temperature regimes. The observed trends align well with established findings in the chemical kinetics literature. In conclusion, the proposed CGD-ABC-BP framework offers a highly accurate and interpretable data-driven approach for modeling IDT in complex aviation fuels, and it shows promising potential for practical engineering deployment. Full article