The fractional differential equations are being used to formulate several problems in science and engineering nowadays. This paper introduces the Caputo fractional-order delay differential equations (FODDEs). In the present scheme, FODDEs have multi-term integer- and fractional-order derivatives for delayed or non-delayed terms. These equations are solved by implementing the Chebyshev collocation scheme and discussing its error analysis. The non-linear and linear systems of algebraic equations are obtained after simplifying the above equations. Various numerical examples illustrate the present numerical scheme’s reliability and efficiency, which are compared graphically with the recently proposed methods. The numerical results obtained through this scheme are easy and efficient to implement when solving various kinds of FODDEs. AMS Subject Classification: 34A08, 34K37

This paper presents a comprehensive numerical study on the efficacy of a Hermite polynomial-based least-squares method for solving Volterra–Fredholm fractional integro-differential equations (V-FFIDEs). In our approach, we construct an approximate solution as a finite expansion of Hermite polynomials. This trial solution is systematically substituted into the governing V-FFIDE. Following the analytical evaluation of the fractional and integral operators, we formulate a residual function. The core of our method involves minimizing the squared norm of this residual over the problem domain, a process that transforms the original problem into a well-defined system of linear algebraic equations. To validate our methodology, we conducted a series of numerical experiments on a collection of representative examples. The results of our study, presented through detailed tables of numerical outcomes and comparative graphical illustrations, conclusively demonstrate the high accuracy, computational efficiency, and robust convergence of the proposed technique.

A unique mathematical model of a three-phase-three-phase voltage modulator is proposed - components of an asynchronous generator with a contactless thyristor excitation system of a wind power plant and a program code for computer research is developed. The mathematical model is a system of differential equations of electrical state in phase coordinates, which takes into account the mutual influences of the structural elements of the voltage modulator (two asynchronous machines) and the electromagnetic couplings of the electrical circuits of these machines. The theoretical basis of the mathematical model is the basic electrical engineering methods: nodal potentials and Kirchhoff's laws and explicit numerical methods of integrating differential equations - Runge-Kutta and the Gaussian method for solving a linear system of algebraic equations of electrical state in the basis of potentials of independent nodes. The results of a computer study of the electromagnetic processes of the modulator are presented in the form of calculated dependences of instantaneous values ​​of voltages and currents. Figs.: 13. Refs.: 21 titles.

Purpose The purpose of this study is to develop an efficient and accurate numerical technique for solving nonlinear boundary value problems, with particular focus on the Darcy–Brinkman–Forchheimer equation (DBFE) and the quartic strongly nonlinear heat transfer equation (QSNHTE). Design/methodology/approach The approach is developed by constructing a matrix-based method that employs Pell-Lucas polynomials (PLPs) and utilizes the evenly spaced collocation points. Initially, the solution is formulated in a matrix representation, allowing all terms within the problems to be expressed accordingly. The Pell-Lucas collocation method (PLCM) is established by using the matrix formulation with the evenly spaced collocation points. Through this procedure, the original nonlinear equations are transformed into a system of linear algebraic equations, and solving this system yields the coefficient matrix corresponding to the PLP-based solutions. An error analysis is conducted for both problems. Subsequently, numerical implementations are carried out using MATLAB. Additionally, the L∞ and root mean square error norms are computed for various polynomial degrees and parameter values, providing quantitative validation of the method's accuracy. Findings The numerical results demonstrate that the proposed PLCM provides highly accurate and stable solutions for both DBFE and QSNHTE. The computed error norms decrease significantly with increasing polynomial degree, confirming the convergence and reliability of the method. Comparisons with existing numerical approaches reported in the literature show that the proposed technique achieves competitive accuracy, which is further illustrated through tabulated data and graphical representations. Research limitations/implications The proposed PLCM has the limitation that it is currently formulated and tested for one-dimensional nonlinear problems, and its performance has been demonstrated specifically for the DBFE and QSNHTE models; therefore, further studies are needed to evaluate its applicability and efficiency for higher-dimensional and more complex nonlinear systems. Practical implications The proposed PLCM provides a reliable and computationally efficient tool for solving nonlinear boundary value problems arising in porous media flow and nonlinear heat transfer. Its matrix-based structure and straightforward MATLAB implementation make it suitable for practical engineering applications requiring high accuracy and numerical stability. Social implications By improving the accuracy and stability of numerical simulations for porous media flow and nonlinear heat transfer models, the proposed method may contribute indirectly to more efficient energy systems and environmentally sustainable engineering designs. Originality/value The originality of this work lies in the matrix representation of nonlinear differential operators based on PLPs, enabling the numerical treatment of strongly nonlinear boundary value problems arising from the DBFE and QSNHTE.

This article deals with the analysis of the deformed shape of local stability loss of a reinforced flexible beam, occurring due to constrained expansion during heating. The multiple elastic supports of the beam modeled as an elastic medium, which provides constant resistance to both longitudinal and transverse displacements of the rod. The infinitely long beam divided into a buckling region and an adjacent region under compression. The lengths of these regions are unknown and should be determined during the solution process. A part of the potential energy accumulated during compression is expended on the work of internal forces during bending deformation following the loss of stability. This leads to a reduction in the magnitude of the compressive force in the buckled region. The problem of determining the displacement functions and the critical value of the safe heating temperature is formulated as a system of nonlinear differential equations concerning the deformations in the regions of the flexible beam. The solution obtained using the finite difference method, which transforms a system of differential equations into a system of linear algebraic equations. This system takes the closed form with boundary conditions and transversality conditions. A sufficient number of grid nodes for constructing the difference scheme determined through an iterative procedure that compares two adjacent solutions. The criterion for comparison of solutions is a tuple of areas under the graphs of the sought functions, which are calculated through numerical integration using the trapezoidal rule. The obtained final solution compared with the classical solution to the stability problem of an evenly loaded beam, which does not take into account longitudinal displacements. Additionally, it is contrasted with the known solution in the field of operation of continuously welded railway tracks, which also disregards resistance to longitudinal displacements in the buckled region. The refined results obtained by the proposed modified method for calculation of the parameters of the deformed shape of a flexible beam is important for monitoring the pre-critical state of the modelling system.

A new version of the linear integral representation method is developed for solving inverse problems in geophysics. This approach is applied to the interpretation of anomalous time-dependent field data. The reconstruction of field elements is reduced to solving a system of linear algebraic equations (SLAE) with an approximately given right-hand side. Since the matrix elements of this system are derived analytically, the modeling process is significantly simplified. The article also analyzes how the approximation quality of a non-stationary field element depends on the observation network geometry, enabling its optimization for more accurate detection of geological properties. The proposed method for solving inverse problems for hyperbolic partial differential equations with constant coefficients can also be applied to data described by systems of nonlinear PDEs, provided the target field is represented as a composition of components differing in magnitude. Finally, the results of non-stationary gravity field modeling are presented.

One of the most important tasks of modern physical geodesy is to determine a high-frequency part of the Earth’s gravity field (EGF), which inevitably depends on the individual characteristics of a particular computational area. The author investigates the practical aspects of a relatively new theory for modeling the said part of the EGF in a local area, using a series of special spherical functions ensuring local orthogonality. With the global gravitational model EGM2008 as input information, we aim to construct a local analytical one for the high-frequency component of the EGF in the Republic of Burundi territory. To overcome the main technical difficulty, the problem of solving rank-deficient systems of linear algebraic equations (SLAE), special regularization techniques are employed. The developed algorithm generalizes the classical global spherical harmonic analysis (GSHA) and is treated as one of the preferred alternative methods of regional modeling. It enables representing fields with high spatial resolution using a much smaller number of parameters rather than the classical GSHA. Compared with other possible methods of regional analysis, the advantage of the LSGA is ultimate provision of a solution, not only analytical, but also meeting the Laplace differential equation, which, as we know, is fundamentally important in many problems of geophysics in general and physical geodesy in particular

The influence of surface elasticity on the stress-strain state of a crack type III, which occurs under antiplane shear deformations of a linearly elastic body, is investigated. Me-chanical effects that arise near surfaces, particularly at the crack faces, are taken into ac-count using the Gurtin and Murdoch continuum surface-boundary model. Equilibrium condi-tions on the crack surface are formulated, as well as the relationship between surface and body stresses. Using these relationships, refined boundary conditions are written on the upper and lower faces of the crack, which are further analyzed using the methods of the theory of complex variable functions. As a result of this analysis, a first-order singular integro-differential equation with a Cauchy-type kernel is formulated. For its solution, the representa-tion of unknown functions in terms of Chebyshev polynomials of the first kind and the method of collocation on the nodes of these polynomials are used. The solution of the resulting system of linear algebraic equations allows to obtain the coefficients of the specified expansions. A formula for calculating the stress on the crack extension is found, which is expressed by an integral with a Cauchy type kernel. A comprehensive analysis of the peculiarities of the nu-merical implementation of the developed algorithm is carried out. It includes variations in the number of components in the expansions of unknown functions in Chebyshev polynomials and the number of nodes in Gauss quadrature formulas for calculating the specified integral. The behavior of the stress difference between the upper and lower crack faces as well as the dis-tribution of another stress component on the crack extension is graphically illustrated in the vicinity of the right tip. The dependence of these quantities on the values of the uniform shear stress specified on the crack edges is also illustrated. It is shown that the consideration of sur-face elasticity becomes especially noticeable when the crack length is less than a micrometer. Further decrease of this length leads to significant change of the character of the stress dis-tribution in the vicinity of the crack tip. In particular, the square root singularity of the stresses at the crack tips, which is characteristic for the classical crack model, disappears and the stresses at these tips become finite.

The key properties of numerical methods for solving ordinary differential equations are determined by their accuracy and stability. The step size is selected based on the accuracy of the numerical solution. In this paper, we will consider difference methods for the approximate solution of first-order ordinary differential equations. Here, we will find the square of the norm of the error functional of difference formulas. To obtain optimal coefficients, we will construct and analyze systems of linear algebraic equations. By solving this system, we will find the optimal coefficients of the difference formulas for specific spaces, and here we will calculate the square of the norm of the error functionals of the optimal difference formulas.

Antisymmetric forms (A-A) of the stability loss of a highly elastic rectangular plate in which two parallel facesare pinched, and the other two are free (CFCF), under the influence of a compressive load on the pinched faces,are investigated. The desired shapes are represented by two odd hyperbolic-trigonometric series with coefficientswhich should ensure the exact fulfillment of all the conditions of the problem. The problem was reduced to solving ahomogeneous infinite system of linear algebraic equations with respect to a single sequence of coefficients containingas a parameter the desired critical load which was found by “firing” during the iterative process. The first three criticalloads for a square plate are found and their 3D images are presented. The results obtained can be used in calculationsof sensitive elements of various sensors in microelectronics, biology, and medicine.

ABSTRACT Loaded differential equations are crucial for representing occurrences having multiple pointwise effects on the overall state. This article presents an analytical as well as numerical study on the spatially loaded Caputo‐type time‐fractional diffusion equation with initial and non‐homogeneous Dirichlet boundary conditions. The well‐posedness of the system is established using the Galerkin approximation method, and some a priori estimates of the state variable are derived. The numerical discretization of the problem is achieved via the finite difference method, which uses the well‐known method for Caputo derivative approximation. This process yields a spatially loaded implicit finite difference scheme, which reduces to a parametric system of linear equations. There, the superposition property of systems of linear algebraic equations is used to obtain a parametric representation of solutions using auxiliary linear systems of special parametric structures. In subsequent analysis, the unique solvability of the proposed scheme is proved. Also, the discrete energy method is implemented to derive unconditional stability and convergence of the proposed scheme for non‐positive load coefficients. In the end, several numerical experiments are conducted over a few test examples to validate the accuracy and efficiency of the proposed scheme.

A rigorous electromagnetic approach has been developed for the theoretical analysis of plasmon resonance excitation frequencies in a two-dimensionally confined structure consisting of a graphene square separated from a homogeneous graphene plane by a dielectric layer. The method is based on the formation of a system of integral equations for the components of the oscillating current in the graphene square. The integral equations are solved using the Galerkin method, which transforms the integral equations into a system of linear algebraic equations. Terahertz spectra of the plasmon resonance absorption cross sections in this structure are constructed. The excitation frequencies of "acoustic" and "optical" two-dimensional plasmons in this structure are found. Cases of a dielectric layer thin and thick compared to the plasmon wavelength are studied. Resonances corresponding to the excitation of mixed-parity plasmonsare detected.

This article discusses the application of the method of integral equations for the numerical evaluation of sound scattering characteristics by a finite elastic cylindrical shell placed in a liquid medium and separated by one or more elastic partitions. The shell has hemispherical ends and is also filled with a liquid medium. The scattered sound field is determined based on the combined use of Kirchhoff-type integral equations and an integral equation for the displacement vector of an elastic medium obeying the Lame equation. Boundary conditions for stresses and displacements are formulated for each of the shell's contact surfaces with the external and internal media. The approach under consideration is based on the numerical transformation of continuous integral equations into a system of linear algebraic equations using curved isoparametric boundary elements. Based on the numerical results, a comparative assessment of the effect of the partition on the scattered sound field for various wave sizes of the shell and location angles was obtained.

We analyse classical and temporally loaded diffusion equations on a metric star graph and discuss possible real-world aspects. We establish the existence and uniqueness of the temporally loaded diffusion equation on a metric star graph using eigenfunction expansions and obtain some apriori estimates. Numerical schemes for considered problems are derived by finite difference approximations. The transmission conditions impose challenges by developing an additional difference scheme at the junction. Alongside, the approximation of loaded diffusion equation leads to loaded difference schemes, and appropriate solution is appointed using the superposition property of a system of linear algebraic equations to deal with loaded terms in the scheme. Thereafter, the unique solvability of difference schemes is derived. The unconditional stability of proposed schemes for classical and temporally loaded diffusion equations on metric star graphs is proved via the discrete energy method. The schemes are convergent with significant rate of O ( Δ t + max 1 ≤ i ≤ k Δ x i 2 ) . The consideration of metric star graph for these problems is unprecedented and approaches used for solving loaded diffusion equation and in subsequent analyses are novel. The paper concludes theoretical findings of proposed schemes by showcasing compatible numerical results from several experiments over test problems results from several experiments over test problems.

A unified analytic solution framework for buckling analysis of single-edge-cracked rectangular plates

Abstract A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand–Levitan equation and the Fourier–Legendre series expansion of the transmutation kernel. A linear algebraic system of equations is obtained, which can be solved numerically. To the best of our knowledge, this is the first practical method for the solution of the inverse problem for the one-dimensional Dirac equation on a finite interval.

This paper presents a solution to the plane doubly periodic problem of loading an infinite, isotropic, elastic plane with a hexagonal packing of elliptical inclusions. The plane is subjected to one of three types of remote loading: uniaxial tension along one of the inclusion’s axes or pure shear. The regular hexagonal unit cell contains a single elliptical inclusion with its axes perpendicular to the cell boundaries. The size of the inclusion is significantly larger than the plate thickness. The solution is constructed by reducing the problem to finding complex potentials from the boundary conditions, which are derived from the equality of normal forces and displacements in the matrix and the inclusion. This is achieved using conformal mapping techniques and the method of Muskhelishvili. The influence of non-central inclusions is accounted for via a small parameter method. As a result, a system of linear algebraic equations is derived for solving the considered doubly periodic problem, and solutions for several specific cases are obtained. The results were verified against a numerical solution obtained by the finite element method in the Abaqus software package. The solution to this problem serves as a model for the loading of a fibrous composite, which makes it highly relevant. A relatively small number of works in mechanics are devoted to fibrous composites, with most publications focusing on the analysis of experimental studies or numerical solutions. Therefore, this analytical solution possesses significant scientific value.

In engineering design, structural defects such as cracks can lead to catastrophic failures in machines and equipment. This study investigates the uniaxial stress state of an infinite sheet weakened by two elliptical holes with linear cracks. Due to the complexity of the geometric configuration and the absence of known conformal mapping functions for such regions, this problem has not been addressed in previous research. We solve a plane elasticity problem for a complex geometric configuration featuring two elliptical holes with linear cuts using the theory of complex variables and conformal mapping functions. The solution involves solving a system of linear algebraic equations derived from the theory of complex variables and Kolosov-Muskhelishvili potentials. By expanding the functions φ(z) and ψ(z) into series, we obtain an analytical solution and provide numerical examples to illustrate key theoretical aspects. The coefficients of the analytical functions are determined, and well-known elasticity theory formulas are applied to compute stress components at characteristic points. This research presents a novel approach to solving this specific problem, as conformal mapping functions for such complex configurations have not been previously established.

Background: This study presents a useful new framework that uses Bernstein polynomials to improve a spectral collocation technique for numerically solving Fredholm integro-differential equations of fractional orders with variable coefficients and multi-time constant delay (FIFDEs-Delays) under boundary conditions. Materials and Methods: The approximate solutions are assumed to be in the form of the truncated Bernstein polynomial series. This novel approach is based on the use of a matrix technique to convert the display equation with conditions into an algebraic linear system of equations with unknown Bernstein coefficients. Results: This approach improves the accuracy of the solutions found while simultaneously simplifying the problem. The solution of this system determines the coefficients of the assumed solution. In addition, the integral operators employed in this technique were quantitatively evaluated using the Clenshaw-Curtis formula. Conclusion: we provide specific examples to showcase the accuracy of the method, and we employ the least-squares error methodology to minimize error terms within the given domain. Ultimately, the most common application suggested for the numerical approaches is implemented in a Python program.

Within the framework of the theory of microstructural deformation, a new approach is proposed for constructing a solution to the bending equation of a long rectangular nanoplate that is under the influence of a transverse load. The proposed approach is based on the collocation method using a system of orthogonal Chebyshev polynomials of the first kind. The bending function is represented as a partial sum of a series of these polynomials. The roots of Chebyshev polynomials of the first kind are chosen as the collocation points. By sequentially multiplying the left and right sides of the resulting matrix equation by the inverse matrix to the matrix with the values of Chebyshev polynomials at the collocation points and by the generalized inverse matrix to the degenerate matrix of differentiation of these polynomials, the equation of the bending surface, taking into account boundary conditions, is reduced to a system of linear algebraic equations with respect to unknown coefficients in the representation of the solution. In this case, the elements of each of these matrices are presented explicitly. An estimate of the error of the constructed solution based on an infinite norm is obtained. The results of the conducted computational experiments are presented, which demonstrate the effectiveness of the proposed approach.