Search Results (59)

Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed based on the expansion formula and parameter-dependent discretizations of the second derivative. Examples illustrating the application of these approximations to the numerical solution of ordinary and partial fractional differential equations are presented, and the convergence and order of the difference schemes are proved. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy of the numerical methods. Full article

This study presents a novel computational framework for approximating solutions to time-fractional partial differential equations (TFPDEs) with variable coefficients, employing the Caputo definition of fractional derivatives. TFPDEs, distinguished by their fractional-order time derivatives, inherently capture the non-local and memory-dependent dynamics observed in a wide range of physical and engineering systems. The proposed method reformulates the TFPDE into a set of decoupled fractional-order ordinary differential equations (FODEs) via Fourier expansion strategy. This decomposition facilitates analytical tractability while preserving the essential features of the original system. The initial conditions of each resulting FODE are systematically obtained from the governing equation’s initial data. Auxiliary initial value problems are formulated for each FODE to facilitate the construction of explicit particular solutions. These solutions are then synthesized through a carefully designed linear superposition, optimized to minimize the residual error across the domain of interest. This residual minimization ensures that the composite solution closely approximates the behavior of the original TFPDE, offering both accuracy and computational efficiency. Theoretical analysis demonstrates that the method is convergent. A FLOP-based analysis confirms that the proposed method is computationally efficient. The validity and effectiveness of the proposed scheme are demonstrated through a set of benchmark problems. Empirical convergence rates are compared with those from existing numerical methods in each case. The findings confirm that the proposed approach consistently achieves superior accuracy and demonstrates robust performance under a wide range of scenarios. These findings highlight the method’s potential as a powerful and versatile tool for solving complex TFPDEs in mathematical modeling and applied sciences. Full article

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived from generalized shifted Jacobi (GSJ) polynomials. These basis functions are uniquely formulated to satisfy the homogeneous form of RBCs (HRBCs). Key to this approach is the establishment of operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) of the constructed polynomials. The application of this framework effectively reduces the given FDE and its RBC to a system of nonlinear algebraic equations that are solvable by standard numerical routines. We provide theoretical assurances of the algorithm’s efficacy by establishing its convergence and conducting an error analysis. Finally, the efficacy of the proposed algorithm is demonstrated through three problems, with our APPSs compared against exact solutions (ExaSs) and existing results by other methods. The results confirm the high accuracy and efficiency of the scheme. Full article

In some problems, partial differential equations are reduced to ordinary differential equations. In special cases, when incorporating randomness, equations can be reduced to systems of stochastic differential Equations (SDEs). Stochastic averaging for a class of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise is considered. Stability criteria for systems of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise do not currently exist. Usually, studies on determining the sensitivity of solutions to the accuracy of setting the initial conditions are being conducted to explain the phenomenon of deterministic chaos. These studies show both convergence in mean square and convergence in probability to averaged systems of stochastic differential equations driven by fractional Brownian motion and Lévy process. The solutions to systems can be approximated by solutions to averaged stochastic differential equations by using the stochastic averaging. Full article

The main goal of this paper is to present a new numerical algorithm for solving two models of one-dimensional fractional partial differential equations (FPDEs) subject to initial conditions (ICs) and integral boundary conditions (IBCs). This paper builds a modified shifted Chebyshev polynomial of the second kind (MSC2Ps) basis function that meets homogeneous IBCs, named IMSC2Ps. We also introduce two types of MSC2Ps that satisfy the given ICs. We create two operational matrices (OMs) for both ordinary derivatives (ODs) and Caputo fractional derivatives (CFDs) connected to these basis functions. By employing the spectral collocation method (SCM), we convert the FPDEs into a system of algebraic equations, which can be solved using any suitable numerical solvers. We validate the efficacy of our approach through convergence and error analyses, supported by numerical examples that demonstrate the method’s accuracy and effectiveness. Comparisons with existing methodologies further illustrate the advantages of our proposed technique, showcasing its high accuracy in approximating solutions. Full article

This study explores the existence and Mittag–Leffler stability of solutions for fuzzy fractional systems that include Caputo derivatives and ordinary derivatives with non-local conditions using the Schauder fixed-point theorem. Following this, we employ the Laplace transform method and numerical techniques to create iterative methods for obtaining exact and approximate solutions. Full article

The COVID-19 pandemic has underscored the importance of understanding the interplay between the cardiovascular and immune systems during viral infections. SARS-CoV-2 enters human cells via the ACE-2 enzyme, initiating a cascade of immune responses. This study presents a coupled mathematical model that integrates the cardiovascular system (CVS) and immune system (IS), capturing their complex interactions during infection. The CVS model, based on ordinary differential equations, describes heart dynamics and pulmonary and systemic circulation, while the IS model simulates immune responses to SARS-CoV-2, including immune cell interactions and cytokine production. A coupling strategy transfers information from the IS to the CVS at specific intervals, enabling the exploration of immune-driven cardiovascular effects. Numerical simulations examined how these interactions influence infection severity and recovery. The coupled model accurately replicated the evolution of cardiac function in survivors and non-survivors of COVID-19. Survivors exhibited a left ventricular ejection fraction (LVEF) reduction of up to $25\%$ while remaining within normal limits, whereas non-survivors showed a severe 4-fold decline, indicative of myocardial dysfunction. Similarly, the right ventricular ejection fraction (RV EF) decreased by approximately $50\%$ in survivors but underwent a drastic 5-fold reduction in non-survivors. These findings highlight the model’s capacity to distinguish differential cardiac dysfunction across clinical outcomes and its potential to enhance our understanding of COVID-19 pathophysiology. Full article

Bioconvection phenomena play a pivotal role in diverse applications, including the synthesis of biological polymers and advancements in renewable energy technologies. This study develops a comprehensive mathematical model to examine the effects of key parameters, such as the Lewis number (Lb), Peclet number (Pe), volume fraction ($\phi$), and angle of inclination $\left(\alpha \right)$, on the flow and heat transfer characteristics of a nanofluid over an inclined cylinder embedded in a non-Darcy porous medium. The investigated nanofluid comprises nano-encapsulated phase-change materials (NEPCMs) dispersed in water, offering enhanced thermal performance. The governing non-linear partial differential equations are transformed into dimensionless ordinary differential equations using similarity transformations and solved numerically via the Network Simulation Method (NSM) and an implicit Runge–Kutta method implemented through the bvp4c routine in MATLAB R2021a. Validation against the existing literature confirms the accuracy and reliability of the numerical approach, with strong convergence observed. Quantitative analysis reveals that an increase in the Peclet number reduces the shear stress at the cylinder wall by up to 18% while simultaneously enhancing heat transfer by approximately 12%. Similarly, the angle of inclination ($\alpha$) significantly boosts heat transmission rates. Additionally, higher Peclet and Lewis numbers, along with greater nanoparticle volume fractions, amplify the density gradient of microorganisms, intensifying the bioconvection process by nearly 15%. These findings underscore the critical interplay between bioconvection and transport phenomena, providing a framework for optimizing bioconvection-driven heat and mass transfer systems. The insights from this investigation hold substantial implications for industrial processes and renewable energy technologies, paving the way for improved efficiency in applications such as thermal energy storage and advanced cooling systems. Full article

This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature. Full article

The paper studies a nonlinear equation including both fractional and ordinary derivatives. The singular Cauchy problem is considered. The theorem of the existence of uniqueness of a solution in the neighborhood of a fixed singular point of algebraic type is proved. An analytical approximate solution is built, an a priori estimate is obtained. A formula for calculating the area where the proven theorem works is obtained. The theoretical results are confirmed by a numerical experiment in both digital and graphical versions. The technology of optimizing an a priori error using an a posteriori error is demonstrated. Full article

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper. Full article

This research, with its potential to revolutionise the construction industry, aims to develop quaternary-blended composites (QBC) by replacing 80% of ordinary Portland cement (OPC) with metakaolin, rice husk ash, and wood ash combined with discrete hybrid natural fibres at a volume fraction of 0.5%. This study investigates the mechanical properties, including compressive strength, split tensile strength, and impact strength of the QBC at various curing ages of 7, 28, and 56 days. Scanning electron microscopy (SEM) analysis was performed to assess the microstructural characteristics. This research aimed to formulate a novel quaternary binder that may minimise our reliance on cement. The experimental results indicate that the mix labelled M4L2 exhibited superior compressive and split tensile strength performance, with percentage increases of approximately 51.03% and 29.19%, respectively. Meanwhile, the M5L1 mix demonstrated enhanced impact energy, with a percentage increase of about 36.40% in 56 days. SEM observations revealed that the MC4 mix contained unhydrated portions and larger cracks. In contrast, the presence of fibres in the M4L2 mix contributed to crack resistance, resulting in a denser matrix and improved microstructural properties. This study also employed an artificial neural network (ANN) model to predict the compressive, tensile, and impact strength characteristics of the QBC, with the predictions aligning closely with the experimental results. An investigation was conducted to determine the ideal number of hidden layers and neurons in each layer. The model’s effectiveness was evaluated using statistical metrics such as correlation coefficient (R), coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MEA), and mean absolute percentage error (MAPE). The findings suggest that the developed QBCs can effectively reduce reliance on conventional cement while offering improved mechanical properties suitable for sustainable construction practices. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

In this work, we study the univariate quantitative smooth approximations, including both real and complex and ordinary and fractional approximations, under different functions. The approximators presented here are neural network operators activated by Richard’s curve, a parametrized form of logistic sigmoid function. All domains used are obtained from the whole real line. The neural network operators used here are of the quasi-interpolation type: basic ones, Kantorovich-type ones, and those of the quadrature type. We provide pointwise and uniform approximations with rates. We finish with their applications. Full article

Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically $\alpha$, significantly influencing the accuracy of the outcome and the associated computational cost. Full article