Search Results (49)

This article investigates the influence of wall permeability on channel flows and addresses the lack of studies that quantify entropy generation in magnetized Casson fluid models using wavelet-based numerical schemes. We introduce a Fibonacci Wavelet Collocation Method (FWCM) to efficiently solve the transformed nonlinear ordinary differential equations and demonstrate its applicability to the coupled momentum and energy equations. The analysis includes detailed graphical and numerical evaluations of entropy generation, temperature, and velocity fields, along with the Bejan number, Nusselt number, and skin-friction variations. The results reveal that entropy generation increases by approximately 18–22% with a higher Biot number and by nearly 15% with increasing Grashof number, while it decreases by about 12% for higher Eckert numbers. Magnetic field strength exhibits a dual effect, producing both suppressing and enhancing behaviors depending on parameter ranges. The FWCM solutions show strong agreement with previously published data, confirming both accuracy and robustness. Full article

Advection–diffusion–reaction-type interface models have wide-ranging applications in environmental science, chemical engineering, and biological systems, particularly in modeling pollutant transport in groundwater, reactive flows, and drug diffusion across biological membranes. This paper presents a novel numerical method for the solution of these models. The proposed method integrates the meshless collocation technique with the finite difference method. The temporal derivative is approximated using a finite difference scheme, while spatial derivatives are approximated using radial basis functions. The interface across the fixed boundary is treated with discontinuous diffusion, advection, and reaction coefficients. The proposed numerical scheme is applied to both linear and non-linear models. The Gauss elimination method is used for the linear models, while the quasi-Newton linearization method is employed to address the non-linearity in non-linear cases. The $L_{\infty }$ error is computed for varying numbers of collocation points to assess the method’s accuracy. Furthermore, the performance of the method is compared with the Haar wavelet collocation method and the immersed interface method. Numerical results demonstrate that the proposed approach is more efficient, accurate, and easier to implement than existing methods. The technique is implemented in MATLAB R2024b software. Full article

This paper presents a novel B-spline wavelet-based scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. By combining semi-orthogonal B-spline wavelets with a collocation approach and a quasilinearization technique, we transform the original problem into a system of algebraic equations. To enhance the computational efficiency, we derive the operational matrix formulation of the proposed scheme. We provide a rigorous convergence analysis of the method and demonstrate its accuracy and effectiveness through numerical experiments. The results confirm the robustness and computational advantages of our approach for solving this class of fractional differential equations. Full article

This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate $R_c\left(M\right)$, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision. Full article

The atmospheric boundary layer (ABL) and tropopause play critical roles in weather formation and climate change. This study initially focuses on the ABL height (ABLH), tropopause height (TPH), and temperature (TPT) retrieved from the integrated radio occultation (RO) profiles from FY-3E, FY-3F, and FY-3G satellites during September 2022 to August 2024. All three FY-3 series satellites are equipped with the RO payload of Global Navigation Satellite System Radio Occultation Sounder-II (GNOS-II), which includes open-loop tracking RO observations from the BeiDou navigation satellite system (BDS) and the Global Positioning System (GPS). The wavelet covariance transform method was used to determine the ABL top, and the temperature lapse rate was applied to judge the tropopause. Results show that the maximum ABL detection rate of FY-3/GNOS-II RO can reach up to 76% in the subtropical eastern Pacific, southern hemisphere Atlantic, and eastern Indian Ocean. The ABLH is highly consistent with the collocated radiosonde observations and presents distinct seasonal variations. The TPH retrieved from FY-3/GNOS-II RO profiles is in agreement with the radiosonde-derived TPH, and both TPH and TPT from RO profiles display well-defined spatial structures. From 45°S to 45°N and south of 55°S, the annual cycle of the TPT is negatively correlated with the TPH. This study substantiates the promising performance of FY-3/GNOS-II RO measurements in observing the ABL and tropopause, which can be incorporated into the weather and climate systems. Full article

The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We investigated the existence, uniqueness, and numerical stability of LDG-based discretized formulation, leveraging the Liouville–Caputo fractional derivative and upwind numerical fluxes to discretize governing equations while preserving stability. The method was validated through benchmark test cases, including comparisons with analytical solutions and established numerical techniques (e.g., Gegenbauer wavelets and Dickson collocation). The results demonstrate that the LDG method achieves high-accuracy solutions (e.g., with a relatively large time step size) and reduced computational costs, which are attributed to its element-wise formulation. These findings position LDG as a promising tool for complex scientific and engineering applications, particularly in modeling fractional-order systems such as RL, RLC circuits, and other electrical circuit equations. Full article

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article

In this paper, neutral delay differential equations, which contain constant and proportional terms, termed mixed neutral delay differential equations, are solved numerically. Moreover, an efficient numerical approach is introduced (a combination of the method of steps and the Haar wavelet collocation method) to solve mixed neutral delay differential equations. Furthermore, we prove the existence and uniqueness theorem using successive approximation methods. Three numerical examples are presented to demonstrate the implementation of the proposed method. Furthermore, the precision and accuracy of the Haar wavelet collocation method are validated theoretically by proving that the error tends to zero as the resolution level increases, and numerically, by calculating the rate of convergence. The findings contribute to a broader application of the wavelet-based method to a more complex type of differential equation. This study offers a framework for the extension of the combination of both methods to be applied to potential real-world applications in control theory, biological models, and computational sciences. Full article

In this paper, we introduce the Hermite wavelet method (HWM), a numerical method for the fractional-order Bagley–Torvik equation (BTE) solution. The recommended method is based on a polynomial called the Hermite polynomial. This method uses collocation points to turn the given differential equation into an algebraic equation system. We can find the values of the unknown constants after solving the system of equations using the Maple program. The required approximation of the answer was obtained by entering the numerical values of the unknown constants. The approximate solution for the given fractional-order differential equation is also shown graphically and numerically. The suggested method yields straightforward results that closely match the precise solution. The proposed methodology is computationally efficient and produces more accurate findings than earlier numerical approaches. Full article

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique. Full article

The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency. Full article

Our goal in this work is to solve the fractional Bratu equation, where the fractional derivative is of the Caputo type. As we know, the nonlinearity and derivative of the fractional type are two challenging subjects in solving various equations. In this paper, two approaches based on the collocation method using Müntz–Legendre wavelets are introduced and implemented to solve the desired equation. Three different types of collocation points are utilized, including Legendre and Chebyshev nodes, as well as uniform meshes. According to the experimental observations, we can confirm that the presented schemes efficiently solve the equation and yield superior results compared to other existing methods. Also, the schemes are convergent. Full article

In the present paper, we consider an effective computational method to analyze a coupled dynamical system with Caputo–Fabrizio fractional derivative. The method is based on expanding the approximate solution into a symmetry Haar wavelet basis. The Haar wavelet coefficients are obtained by using the collocation points to solve an algebraic system of equations in mathematical physics. The error analysis of this method is characterized by a good convergence rate. Finally, some numerical examples are presented to prove the accuracy and effectiveness of this method. Full article

A hybrid efficient and highly accurate spectral matrix technique is adapted for numerical treatments of a class of two-pint boundary value problems (BVPs) with singularity and strong nonlinearity. The underlying model is a reaction-diffusion equation arising in the modeling of biomedical, chemical, and physical applications based on the assumptions of Michaelis–Menten kinetics for enzymatic reactions. The manuscript presents a highly computational spectral collocation algorithm for the model combined with the quasilinearization method (QLM) to make the proposed technique more efficient than the corresponding direct spectral collocation algorithm. A novel class of polynomials introduced by S.K. Chatterjea is used in the spectral method. A detailed proof of the convergence analysis of the Chatterjea polynomials (ChPs) is given in the $L_2$ norm. Different numerical examples for substrate concentrations with all values of parameters are performed for the case of planar and spherical shapes of enzymes. For validation, these results are compared with those obtained via wavelet-based procedures and the Adomian decomposition scheme. To further improve the approximate solutions obtained by the QLM–ChPs method, the technique of error correction is introduced and applied based on the concept of residual error function. Overall, the presented results with exponential convergence indicate that the QLM–ChPs algorithm is simple and flexible enough to be applicable in solving many similar problems in science and engineering. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article