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This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB. [ABSTRACT FROM AUTHOR]

The aim of this study is to present the evolution of COVID‐19 pandemic in Turkey. For this, the SIR (Susceptible, Infected, Removed) model with the fractional order derivative is employed. By applying the collocation method via the Pell–Lucas polynomials (PLPs) to this model, the approximate solutions of model with fractional order derivative are obtained. Hence, the comments are made about the susceptible population, the infected population, and the recovered population. For the method, firstly, PLPs are expressed in matrix form for a selected number of N$$ N $$. With the help of this matrix relationship, the matrix forms of each term in the SIR model with the fractional order derivative are constituted. For implementation and visualization, we utilize MATLAB. Moreover, the outcomes for the Runge–Kutta method (RKM) are obtained using MATLAB, and these results are compared with the results obtained with the Pell–Lucas collocation method (PLCM). From all simulations, it is concluded that the presented method is effective and reliable. [ABSTRACT FROM AUTHOR]

It is challenging to use the piecewise polynomial collocation method numerically solving the Caputo fractional differential equations (FDEs), since it is related to the well-known Conjecture 6.3.5 in Brunner’s 2004 monograph on the convergence of the collocation solution for weakly singular Volterra integral equations (VIEs) of the first kind, and this is the reason why in the existing literature, the collocation method is not used directly to solve FDEs, but rather indirectly to solve the reformulated VIEs. The Bagley–Torvik equation is a typical representative of a class of FDEs, whose highest order derivative of the unknown function is an integer, and in which a Caputo derivative is also involved, and as the highest-order derivative is of integer order, one can use collocation methods directly to numerically solve the Bagley–Torvik equation. In this paper, the direct piecewise polynomial collocation method is used to solve the Bagley–Torvik equation, and the global convergence is derived on graded meshes and the pointwise error estimate is obtained on uniform meshes. Moreover, the global superconvergence of the collocation solution is also directly obtained without any postprocessing techniques. Unlike the indirect reformulated numerical methods, one has to resort to the iterated numerical solution to improve the numerical accuracy. Some numerical examples are given to illustrate the theoretical results, and it also shows that our analysis for the Bagley–Torvik equation can be extended to more general integer order derivative dominant FDEs, even for time fractional partial differential equation with this characteristic. [ABSTRACT FROM AUTHOR]

Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is m − 2 for Sobolev kernel with kernel smoothness m. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles. [ABSTRACT FROM AUTHOR]

This article introduces a Haar wavelet-based numerical method for solving fifth-order linear and nonlinear differential equations. This method easily handles both homogeneous and nonhomogeneous equations. It also works with variable and constant coefficients under various conditions. The method is flexible, making it easy to work with boundary, integral, and two-point integral conditions. These three different cases of given information are coupled with fifth-order linear and nonlinear differential equations, and the method proves to be effective in these cases. The outcomes of the Haar wavelet collocation technique are compared with approaches found in existing literature. The method demonstrates second-order convergence, and experimental results support this idea as well. The CPU time is used to evaluate the efficiency of the method, and the maximum absolute errors ( L ∞ ) are utilized to assess the accuracy level. Different examples are studied along with various given information, and the method is found to be adaptable to different types of boundary conditions and particular integral conditions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a precise and efficient ultraspherical wavelet method for a famous Benjamin-Bona-Mahony (BBM) mathematical model. The suggested technique uses the collocation method and ultraspherical wavelets. The proposed scheme is applied to linear and nonlinear BBM equations to inspect the efficiency and accuracy of the proposed technique. The effectiveness of this practical approach is verified. Moreover, the method based on the ultraspherical wavelets is simple, accurate, fast, flexible, and convenient. The results are analyzed using tables and graphs and compared with other methods in literature. As we know, many partial differential equations (PDEs) don't have exact solutions, and some semi-analytical methods work based on controlling parameters, but this is a controlling parameter-free technique. Also, it is pretty simple to implement and consumes less time to execute the programs. The recommended wavelet-based numerical approach is interesting, productive, and efficient. The proposed technique's convergence analysis is also presented through the theorem. [ABSTRACT FROM AUTHOR]

This study introduces a local radial basis function collocation method (LRBFCM) to analyzing structural deformation in deep excavation within a two dimensional geotechnical model. To mitigate the size effect caused by a large length-to-width ratio, a technique known as the 'direct method' is employed. This method effectively reduces the influence of the shape parameter, thereby improving the accuracy of the partial derivative calculations in LRBFCM. The combination of LRBFCM with the direct method is applied to the deep excavation problem, which consists of both the soil and support structures. The soil is modeled using the Drucker-Prager (D-P) elastic-plastic model, while an elastic model is employed for the support structure. Elastic-plastic discretization is performed using incremental theory. The proposed approach is validated through four different examples, comparing the results with numerical solutions obtained from traditional finite element methods (FEM). This study advocates the use of the direct method to optimize the distribution of local influence nodes, particularly in cases involving large length-to-width ratios. The combination of LRBFCM with incremental theory is shown to be effective for addressing elastic-plastic problems. [ABSTRACT FROM AUTHOR]

Axisymmetric contact problems on the interaction of a rigid stamp and a poroelastic layer, the bottom boundary of which is located on a Winkler base, were considered based on the equations of the Cowin–Nunziato theory of poroelastic bodies. It was assumed that the base of the cylindrical stamp has the flat or parabolic shape and there is no friction in the contact area. With the help of the Hankel integral transformation, the problems posed were reduced to the integral equations for an unknown contact stresses, which were solved with the use of the collocation methods, while the transformants of the kernels of the integral equations were obtained in explicit form. The values of contact stresses and size of the contact area in the case of the parabolic stamp were determined. The relationship, which is one of the main characteristics, when determining the mechanical parameters of a material by the indentation method, between the force acting on the stamp and its displacement, was studied. A comparative analysis of the studied quantities for various values of the poroelastic parameters and the coefficient of subgrade resistance of the Winkler base was carried out. The model of contact interaction considered and the solution obtained can be used in calculation of the structures foundations located on ground bases. [ABSTRACT FROM AUTHOR]

Thermal convection induced by internal heating appears in different natural situations and technological applications with different internal sources of heat (e.g., radiation, electric or magnetic fields, chemical reactions). Thermal convection due to Joule heating in weak electrical conducting liquids such as molten salts with symmetric thermal boundary conditions is investigated using linear stability analysis. We show that, in the quasi-static approximation where the induced magnetic field is negligible, the effect of the external magnetic field consists of the delay in the threshold of thermal convection and the increase in the size of thermoconvective rolls for an intense magnetic field. Analysis of the budget of the perturbations' kinetic energy reveals that the Lorentz force contributes to the dissipation of the kinetic energy. [ABSTRACT FROM AUTHOR]

The Chebyshev cardinal functions based on the Lobatto grid are introduced and used for the first time to solve the fractional delay differential equations. The presented algorithm is based on the collocation method, which is applied to solve the corresponding Volterra integral equation of the given equation. In the employed method, the derivative and fractional integral operators are expressed in the Chebyshev cardinal functions, which reduce the computational load. The method is characterized by its simplicity, adherence to boundary conditions, and high accuracy. An exact analysis has been provided to demonstrate the convergence of the scheme, and illustrative examples validate our investigation. [ABSTRACT FROM AUTHOR]

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation. [ABSTRACT FROM AUTHOR]

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton's portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton's portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class. In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes. [ABSTRACT FROM AUTHOR]

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances. [ABSTRACT FROM AUTHOR]

In life important applications are modeled mathematically by nonlinear partial differential equations. Primarily, the objective is to transform such equations into a system of algebraic equations to get their solutions. The Legendre collocation method is demonstrated by the differentiation operational matrix via shifted Legendre polynomials in such a transformation. The validity and effectiveness of the prospective method are manifested by illustrative examples, an error analysis, residual analysis, and comparison with others. [ABSTRACT FROM AUTHOR]

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach. [ABSTRACT FROM AUTHOR]

This article presents an SIRD model based on the evolution of Coronavirus Disease 2019 (COVID-19) caused by SARS-CoV-2 from the coronavirus family. Firstly, we constitute Pell-Lucas collocation method (PLCM) for this model. According to method, the matrix forms of the Pell-Lucas polynomials (PLPs) are constituted. By utilizing this matrix forms, solution forms and all terms in this model are expressed in matrix form. Thus, PLCM transforms our model into a system of the matrix equations. By solving this system, the approximate solutions are obtained. In addition, the error analysis is also presented. In the examples of this study, we analyzed the Türkiye's situation using initial datas and the parameters for Türkiye. For this, we make applications for two different scenarios. In these two scenarios, the parameters, the initial conditions and the selected range are different. By considering the initial data and the parameters for other countries, this method can be applied to them, too. Application results are tabulated and visualized. Moreover, by comparing our results with Runge-Kutta method (RKM), the effectiveness of method is demonstrated. This study allows the identification of trends in the pandemic. [ABSTRACT FROM AUTHOR]

Copyright of Afyon Kocatepe University Journal of Science & Engineering / Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi is the property of Afyon Kocatepe University, Faculty of Science & Literature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper a time-fractional Black–Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and 2 - μ is time, where μ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option. Moreover, the classical Black–Scholes model is also incorporated into our numerical study to validate the competence of our method in handling not only fractional problems, but also classical ones with favorable results. [ABSTRACT FROM AUTHOR]

In this work, the Haar wavelet collocation method (HWCM) is used to study the mathematical model of Global warming that impacts the marine ecosystem. The sustainable growth of the environment is only achieved by reducing greenhouse gases (GHGs); the present model describes the harm of GHGs to the environment and marine ecosystem. The considered model is a nonlinear system containing four differential equations (DE) illustrating the concentration of GHGs, atmospheric temperature, planktonic population, and population of fishes. In the current study, we consider the Caputo-Fabrizio (CF) fractional operator to study the present model with the help of HWCM. The operational matrix of integration (OMI) is built using Haar wavelets, which helps us to reduce the present model into a system of algebraic equations by which we can determine the unknown Haar coefficients using Newton Raphson method. To validate the effectiveness of the hired algorithm, we examined the considered system in terms of integer and fractional order. The convergence analysis is presented in the form of theorems. The numerical and graphical comparisons are made between the ND Solver solution, the RK4 solution, and the devised approach. The consequences illustrate that the scheme considered is highly organized and efficient in analyzing the nature of the present model. Numerical computations and implementations are done with the help of Mathematica software. Effects on the ecosystem due to global warming are explained in detail through Graphs and tables. [ABSTRACT FROM AUTHOR]

In this paper, the Sinc-collocation method is applied to solve a system of coupled nonlinear differential equations that report the chemical reaction of carbon dioxide CO2 and phenyl glycidyl ether in solution. The model has Dirichlet and Neumann boundary conditions. The given scheme has transformed this problem into some algebraic equations. The approach is quite simple to handle and the new numerical solutions are compared with some known solutions, which shows that the new technique is accurate and efficient. [ABSTRACT FROM AUTHOR]

In this article, we presented a numerical approach based on non-uniform Haar wavelets to approximate the solution of a second-order singularly perturbed problems with discontinuous data. The solutions to such problems have strong interior layer due to the discontinuity. Accordingly, we have utilized a special type of piecewise uniform mesh called a Shishkin mesh to resolve the layer behaviour of the solution. As the Haar functions are discontinous, the approximate solution is obtained with the integration approach. The second-order derivative is approximated by the linear combinations of Haar functions and then integrated to obtain the numerical approximations. The convergence analysis of the numerical method proposed is carried out, showing that the proposed method is of order two. The adaptability of the proposed method is established by numerical results on two test problems. Even at lower resolution levels, the proposed method provides high accuracy. In any programming language, the proposed method can be easily implemented and is computationally efficient. [ABSTRACT FROM AUTHOR]

In this study, a numerical method based on the Pell-Lucas polynomials (PLPs) is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment. The HIV/AIDS mathematical model with a treatment compartment is divided into five classes, namely, susceptible patients (S), HIV-positive individuals (I), individuals with full-blown AIDS but not receiving ARV treatment (A), individuals being treated (T), and individuals who have changed their sexual habits sufficiently (R). According to the method, by utilizing the PLPs and the collocation points, we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations. Also, the error analysis is presented for the Pell-Lucas approximation method. The aim of this study is to observe the behavior of five populations after days when drug treatment is applied to HIV-infectious and full-blown AIDS people. To demonstrate the usefulness of this method, the applications are made on the numerical example with the help of MATLAB. In addition, four cases of the fractional order derivative () are examined in the range. Owing to applications, we figured out that the outcomes have quite decent errors. Also, we understand that the errors decrease when the value of N increases. The figures in this study are created in MATLAB. The outcomes indicate that the presented method is reasonably sufficient and correct. [ABSTRACT FROM AUTHOR]

A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense is considered and studied through two novel techniques called the Homotopy analysis method (HAM). A reliable approach is proposed for solving fractional order nonlinear ordinary differential equations, and the Haar wavelet technique (HWT) is a numerical approach for both integer and noninteger orders. Perturbation techniques are widely applied to gain analytic approximations of nonlinear equations. However, faturbation methods are essentially based on small physical parameters (called faturbation quantity), but unfortunately, many nonlinear problems have no such kind of small physical parameters at all. HAM overcomes this, and HWT does not require any parameters. Due to this, we opt for HAM and HWT to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solve the models by HAM by choosing the preferred control parameter. Second, HWT is considered. Through this technique, the ofaational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Results on convergence have been discussed in terms of theorems. [ABSTRACT FROM AUTHOR]

A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense is considered and studied through two novel techniques called the Homotopy analysis method (HAM). A reliable approach is proposed for solving fractional order nonlinear ordinary differential equations, and the Haar wavelet technique (HWT) is a numerical approach for both integer and noninteger orders. Perturbation techniques are widely applied to gain analytic approximations of nonlinear equations. However, perturbation methods are essentially based on small physical parameters (called perturbation quantity), but unfortunately, many nonlinear problems have no such kind of small physical parameters at all. HAM overcomes this, and HWT does not require any parameters. Due to this, we opt for HAM and HWT to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solve the models by HAM by choosing the preferred control parameter. Second, HWT is considered. Through this technique, the operational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Results on convergence have been discussed in terms of theorems.

The paper proposes a matrix implementation of the collocation method for constructing a solution to Volterra integral equations of the second kind using systems of orthogonal Chebyshev polynomials of the first kind and Legendre polynomials. The integrand in the equations considered in this work is represented as a partial sum of a series for these polynomials. The roots of the Chebyshev and Legendre polynomials are chosen as collocation points. Using matrix and integral transformations, properties of finite sums of products of these polynomials and weight functions at the zeros of the corresponding polynomials with degree equal to the number of nodes, integral equations are reduced to systems of linear algebraic equations for unknown values of the sought functions at these points. As a result, solutions to Volterra integral equations of the second kind are found by polynomial interpolations of the obtained function values at collocation points using inverse matrices, the elements of which are written on the basis of orthogonal relations for these polynomials. In the presented work, the elements of integral matrices are also given in explicit form. Error estimates for the constructed solutions with respect to the infinite norm are obtained. The results of computational experiments are presented, which demonstrate the effectiveness of the collocation method used.

AbstractIn this paper, we present two collocation methods utilizing subdivision schemes and Bernstein polynomials as their basis functions to solve Caputo-type fractional differential equations. These methods transform fractional differential equations into systems of linear equations, which can be solved using various suitable techniques. Notably, these methods are versatile, making them applicable to both boundary value problems and initial value problems. To demonstrate their effectiveness, we applied these methods to three test problems of fractional differential equations, including the Bagley-Torvik equation and fractional oscillation equation. Our results reveal that both methods provide a more accurate approximation to the exact solution when compared to various algorithms found in the existing literature, showcasing their efficiency, accuracy, and consistency.

Abstract The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances.

Abstract In life important applications are modeled mathematically by nonlinear partial differential equations. Primarily, the objective is to transform such equations into a system of algebraic equations to get their solutions. The Legendre collocation method is demonstrated by the differentiation operational matrix via shifted Legendre polynomials in such a transformation. The validity and effectiveness of the prospective method are manifested by illustrative examples, an error analysis, residual analysis, and comparison with others.

Abstract This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

This study aims to solve the inverse problem for determining the heterogeneity of an object. The scattered field was measured outside its boundaries at a set of observation points. Both the radiation source and observation points were assumed to be located outside the object. The scattered field was modeled by solving the direct problem. The inverse problem was solved using a two-step method. Nonlinearities of various types were considered. When introducing the computational grid, the generalized grid method was applied. A numerical method for solving the problem was proposed and implemented. The numerical results obtained illustrate how the problem is solved for specified experimental data.

This study presents an approximate numerical technique for solving time fractional advection-diffusion-reaction predator-prey equations with variable order (VO), where the analyzed fractional derivatives of VO are in the Caputo sense. Results for Ulam–Hyers stability are shown, as well as the existence and uniqueness of solutions. It is suggested to use a numerical approximation based on the shifted second kind of airfoil polynomials to solve the equations under consideration. A fractional derivative operational matrix with VO is derived for shifted airfoil polynomials, which will be used to compute the unknown function. The main equations are transformed into a set of algebraic equations by substituting the aforementioned operational matrix into the equations under consideration and utilizing the properties of the shifted airfoil polynomial along with the collocation points. A numerical solution is obtained by solving the acquired set of algebraic equations. To verify the accuracy and efficiency of the discussed scheme, several illustrative examples have been considered. The results obtained by the proposed method demonstrate the efficiency and superiority of the method compared to other existing methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a numerical approach for solving linear and nonlinear differential equations arising in astrophysics, commonly known as Lane-Emden equations. The proposed method is based on the collocation method and first involves taking the truncated Chelyshkov series of the function in the equation. The computational cost is reduced due to the orthogonality of Chelyshkov polynomials, and the solution of a linear or nonlinear Lane-Emden equation is reduced to solving a system of linear or nonlinear algebraic equations. Several test examples of these types of differential equations, modeling different physical problems with initial and boundary conditions, are solved to demonstrate the reliability of the method. To demonstrate its effectiveness, absolute error tables and graphs are presented, and the numerical results are compared with other methods and exact solutions. It is observed that when the exact solution has a polynomial form, the proposed method proves to be highly accurate and effective.

In this study, we examine the movement of two hard spheres aligned in a straight line within an incompressible couple stress fluid under the impact of the magnetic field. Both objects have distinct shapes and move along an axis connecting their centers with varying velocities. As a first step, an incompressible analytical analysis is performed on a fluid with couple stress properties around an axially symmetric particle. Using the superposition principle, a general solution is developed for couple stress fluid flows over two moving objects. In order to achieve the boundary conditions, the boundary collocation strategy is applied to the surfaces of the two spheres. A set of tables and graphs illustrates numerical estimates of the dimensionless drag forces acting on two spherical objects. In addition, a drop in Hartmann number or an increase in couple stress viscosity will increase the dimensionless drag force on each spherical particle.

Abstract In this paper, the fractional-order Chelyshkov functions (FCHFs) and Riemann-Liouville fractional integrals are utilized to find numerical solutions to fractional delay differential equations, by transforming the problem into a system of algebraic equations with unknown FCHFs coefficients. An error bound of FCHFs approximation is estimated and its convergence is also demonstrated. The effectiveness and accuracy of the presented method are established through several examples. The resulting solution is accurate and agrees with the exact solution, even if the exact solution is not a polynomial. Moreover, comparisons between the obtained numerical results and those recently reported in the literature are shown.

In this paper, the fractional-order Chelyshkov functions (FCHFs) and Riemann-Liouville fractional integrals are utilized to find numerical solutions to fractional delay differential equations, by transforming the problem into a system of algebraic equations with unknown FCHFs coefficients. An error bound of FCHFs approximation is estimated and its convergence is also demonstrated. The effectiveness and accuracy of the presented method are established through several examples. The resulting solution is accurate and agrees with the exact solution, even if the exact solution is not a polynomial. Moreover, comparisons between the obtained numerical results and those recently reported in the literature are shown. [ABSTRACT FROM AUTHOR]

In this study, a wavelet‐based collocation scheme has been introduced for solving the linear and nonlinear Fredholm integral equations as well as the system of linear Fredholm integral equations with weakly singular logarithmic kernel. Initially, Laguerre wavelets have been constructed by dilation and translation of Laguerre polynomials. For the numerical solution of the Fredholm integral equations, all the functions have been approximated with respect to the Laguerre wavelets. Then, the proposed linear and nonlinear Fredholm integral equations reduce to systems of linear and nonlinear algebraic equations by utilizing the function approximations. Furthermore, the error estimation and the convergence analysis of the presented method have been discussed. Moreover, the numerical results of the several experiments have also been presented in both graphical and tabular form to describe the accuracy and efficiency of the approached method, and also, to determine the validity of the presented scheme, the approximate solutions and absolute error values are compared with the results obtained by other existing approaches. [ABSTRACT FROM AUTHOR]

In this article, we impose fractal features onto classical multiquadric (MQ) functions. This generates a novel class of fractal functions, called fractal MQ functions, where the symmetry of the original MQ function with respect to the origin is maintained. This construction requires a suitable extension of the domain and similar partitions on the left side with the same choice of scaling parameters. Smooth fractal MQ functions are proposed to solve initial value problems via a collocation method. Our numerical computations suggest that fractal MQ functions offer higher accuracy and more flexibility for the solutions compared to the existing classical MQ functions. Some approximation results associated with fractal MQ functions are also presented. [ABSTRACT FROM AUTHOR]

This paper outlines a numerical method called the Bernstein operational matrix of derivative (BOMD) of order two and order three with the approach of the Chebyshev collocation technique to solve boundary value problems (BVP). BOMD with suitable collocation points is implemented to solve the BVP using the linear combination of Bernstein polynomials with unknown coefficients to approximate the solutions. The derivatives featured in the problem sets will be approximated by utilizing the matrix. The subsequent examination involves a mathematical analysis of the proposed method, including evaluating its order, absolute error metrics and comparative assessments with alternative methodologies. Four problems involving linear and non-linear equations and systems, along with practical real-world problems, are addressed to assess the reliability of the proposed method. [ABSTRACT FROM AUTHOR]

The interfacial stress between fibers and matrix plays an important role in the durability and damage initiation of carbon fiber reinforced composites. In this study, thermoelastic analysis was performed on a plate containing randomly distributed multiple fibers. Complex stress functions were employed with a semi-numerical method to ensure displacement continuity along the fiber/matrix interface as a boundary condition. The statistical investigation reveals that the stress concentration due to the presence of multiple fibers increases as the fiber density increases, though its deviation decreases. A numerical case study was conducted to discuss the micromechanics of the inherent scatter of material strength. The stress-strength model with Monte-Carlo simulation demonstrated the fracture probability calculation. The uncertainty obtained is partially attributed to the micromechanical stress variation around the fibers. [ABSTRACT FROM AUTHOR]

The main idea of this work is to present a numerical method based on Vieta‐Fibonacci polynomials (VFPs) for finding approximate solutions of fractal‐fractional (FF) pantograph differential equations and a system of differential equations. Although the presented scheme can be applied to any fractional integral, we focus on the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio integrals with due to their privileges. To carry out the method, first, we introduce FF integral operators in the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio senses. Then, by applying the Vieta‐Fibonacci polynomials and their FF integral operators together with the collocation method, the problem becomes reduced to a system of algebraic equations that can be solved by Mathematical software. In the presented scheme, acceptable approximate solutions are achieved by employing only a few number of the basic functions. Moreover, the error analysis of the presented method is investigated. Finally, the accuracy of the presented method is examined through the numerical examples. The proposed scheme is implemented for some famous systems of FF differential equations, such as memristor, which is a fundamental circuit element so called universal charge‐controlled mem‐element, convective fluid motion in rotating cavity, and Lorenz chaotic system. [ABSTRACT FROM AUTHOR]

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion's invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate. [ABSTRACT FROM AUTHOR]

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order 0 < α < 2 . The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage. [ABSTRACT FROM AUTHOR]

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C ([ 0 , T ] × [ 0 , T ]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown. [ABSTRACT FROM AUTHOR]

In this paper, the numerical method for solving a class of generalized fractional advection-diffusion equation (GFADE) is considered. The fractional derivative involving scale and weight factors is imposed for the temporal derivative and is analogous to the Caputo fractional derivative following an integration-after-differentiation composition. It covers many popular fractional derivatives by fixing different weights $ w(t) $ w (t) and scale functions $ z(t) $ z (t) inside. The numerical solution of such GFADE is derived via a collocation method, where conventional Legendre polynomials are implemented. Convergence and error analysis of polynomial expansions are studied theoretically. Numerical examples are considered with different boundary conditions to confirm the theoretical findings. By comparing the above examples with those from existing literature, we find that our proposed numerical method is simple, stable and easy to implement. [ABSTRACT FROM AUTHOR]

The Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation. [ABSTRACT FROM AUTHOR]

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations w h n and v h n of w (· , t n) and Δ w (· , t n) are constructed. The stability of w h n and v h n are proved, and the a priori bounds of ‖ w h n ‖ and ‖ v h n ‖ are established, remaining α -robust as α → 1 - . Then, the error ‖ w (· , t n) - w h n ‖ and ‖ Δ w (· , t n) - v h n ‖ are estimated with α -robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed α . Finally some numerical results are provided to support our theoretical findings. [ABSTRACT FROM AUTHOR]

This study investigates unsteady velocity U w = ξ x / t for a Williamson nanofluid film flowing over a moving surface. This work can be used to outline the effects of an applied angled magnetic-field on liquid film flow, which occurs in numerous real-world solicitations such as coating industries for wire or sheet, labs, painting, and several others. Analyzing williamson nanoliquid film flow over a stretching sheet is the main aim of this investigation. The leading Navier–Stokes models are reduced to third-order nonlinear ODE through similarity transformations that are then undertaken using the Hermite wavelet method (HWM). Both 2-dimensional and axisymmetric film flow circumstances have been analyzed. The moving surface parameter ξ is said to have a limited range for which the solution exists. Specifically, ξ ≤ - 1 / 4 for axisymmetric flow and ξ ≥ - 1 / 2 for two-dimensional flow. Before decreasing to the boundary condition, the velocity climbs until it reaches its maximum. By taking into account the stretching ( ξ > 0 ) and shrinking ( ξ < 0 ) wall conditions, streamlines are also examined for axisymmetric and 2-dimensional flow patterns. [ABSTRACT FROM AUTHOR]

The principal purpose of this investigation is to present an approximate algorithm for solving third-kind Volterra integral equation with nonlinear vanishing delays using mesh-free techniques based on a radial basis function collocation scheme. This technique does not require background approximation cells, and the algorithm is efficient, has good stability and does not require much computer memory. A description of the approach for the proposed equations is provided. The error estimation of the proposed approach is also provided. The accuracy and reliability of the new scheme are demonstrated through numerical examples. Some of these examples have been compared with analytical solutions and moving least squares methods. The numerical results validate their agreement with theoretical error estimates. [ABSTRACT FROM AUTHOR]