Shifted Chebyshev Polynomials Research Articles

Optimization of dynamic control systems using water cycle algorithm

In this paper, path optimization in dynamic control systems was performed using water cycle algorithm. The core principles and assumptions that guide the suggested technique are drawn from nature and are based on observations of the water cycle process and how rivers and streams naturally flow into the sea. The Water Cycle Algorithm has been recognized as a reliable method for dealing with large-scale optimization challenges, largely because of its proficient exploitation and exploration capabilities. This Algorithm does not require the continuity of functions when specifying the issue, unlike many gradient-based approaches. This is achieved through the use of a simple mathematical model that represents the water cycle process. This model is used to simulate the movement of water droplets in a cloud, where each water droplet represents a potential solution to the optimization problem. Although water cycle optimization has been utilized successfully to tackle optimization issues, this is the first time dynamic control optimization has been used. The control problem is switched out for an optimum programming problem in order to find optimal pathways utilizing the Shifted Chebyshev polynomial. The water cycle method was then used to resolve the new issue. The numerical results demonstrate the excellent precision with which this search method can discover solutions.

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

Numerical solution of third-kind Volterra integral equations with proportional delays based on moving least squares collocation method

In this study, we propose a moving least squares approximation with shifted Chebyshev polynomials to solve linear and nonlinear third-kind Volterra delay integral equations (VDIEs). The suggested approach does not use meshing and does not rely on the geometry of the domain; therefore, we may consider it as a meshless method. This method approximates the solution using the collocation method based on the moving least squares approximation. The formulation of the technique for the suggested equations is described, and its convergence is analysed. Numerical results are presented to demonstrate the high resolution of the proposed approach and confirm its capability to provide accurate and efficient computations for Volterra delay integral equations of the third kind.

Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model

Abstract This study investigates the complex dynamics of a viscoelastic fluid subjected to magneto-hydrodynamics over a stretching sheet, incorporating the Cattaneo–Christov heat flux model. This model is especially advantageous for explaining heat transfer in materials possessing significant thermal conductivity, where the conventional Fourier’s law might not be precise. The investigation revolves around evaluating how the thermal relaxation time affects the boundary layer and how both thermal radiation and viscous dissipation influence the thermal field. The significance of this research lies in its contribution to understanding the intricate behavior of such fluids in the presence of magnetic fields and non-Fourier heat conduction. The primary objective is to analyze the impact of viscoelasticity, magnetohydrodynamics, and Cattaneo–Christov heat flux on the flow and heat transfer characteristics over the stretching sheet. The research methodology involves the application of mathematical models and numerical techniques, particularly the use of the shifted Chebyshev polynomials of the sixth-order approximation and spectral collocation technique. The major conclusion of the study underscores the significant influence of viscoelasticity, magnetic field, and Cattaneo–Christov heat flux on the transport properties of the fluid, providing valuable insights for applications in various engineering and industrial contexts. Certain notable results arising from the current issue indicate that heat transfer is more pronounced for the viscoelastic factor and magnetic parameter, whereas the thermal relaxation parameter exhibits the opposite trend. In addition, the inclusion of the Cattaneo–Christov term enhances our comprehension of thermal behavior.

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given.

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are utilized to model diverse physical phenomena. A matrix collocation method is proposed for determining the approximate solution of this functional equation category. The method employs shifted Chebyshev polynomials of the fifth kind (SCPFK) to construct two-dimensional pseudo-operational matrices of integration, avoiding the need for explicit integration and thereby speeding up computations. Error bounds are examined in a Chebyshev-weighted space, providing insights into approximation accuracy. The approach is applied to several experimental examples, and the results are compared with those obtained using the Bernoulli wavelets and Legendre wavelets methods.

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

AbstractThis work presents a numerical approach for solving the time–space fractional‐order wave equation. The time‐fractional derivative is described in the conformal sense, whereas the space‐fractional derivative is given in the Caputo sense. The investigated technique is based on the third‐kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third‐kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non‐standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.

Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium

Abstract The aim of the present research is to discuss the numerical aspects of heat-mass transfer in power-law nanofluids on an stretched surface. In addition, the novelty in this research lies in its thorough exploration and incorporation of parameters such as viscous dissipation, slip velocity, and convective boundary conditions into the analysis. This distinguishes the study from previous work and underscores its originality. For non-Newtonian fluids, a power-law model is employed, while the nanofluid system associate the influences of thermophoresis and the Brownian motion. The fluid’s thermal conductivity is considered to change based on temperature, while the concentration of nanoparticles at the surface is maintained at a constant level. A heated fluid situated beneath the lower surface can act as a heat convection mechanism source. A process of similarity transformation is employed to simplify the equations related to the mass, momentum, thermal energy, and nanoparticle concentration into nonlinear ordinary differential equations. These equations are then treated numerically with the help of the shifted Chebyshev polynomials of the sixth order and the spectral collocation method. The proposed technique reduces the existing problem into a system of algebraic equations formulated as a constrained optimization challenge. Subsequently, the optimization technique is applied to determine the unknown coefficients of the series solution. Graphical representations depict the impacts of nanofluid parameters. A quantitative assessment is presented in a tabular format to illustrate a comparison with previously published results for specific scenarios, revealing a notable level of agreement.

Innovative Method for Computing Approximate Solutions of Non-Homogeneous Wave Equations with Generalized Fractional Derivatives

In this work, a well-known non-homogeneous wave equation with temporal fractional derivative is approximately investigated. A recently defined generalized non-local fractional derivative is utilized as the fractional operator. A novel technique is proposed to approximate the solutions of wave equation with generalized fractional derivative. The proposed method is based on the shifted Chebyshev polynomials and a combination of collocation and residual function methods. Theoretical analysis of the convergence of the proposed method is performed. Approximate solutions are derived in both rectangular and non-rectangular (general) domains.

Numerical treatment via the spectral collocation method for Casson–Williamson nanofluid flow due to a stretching sheet with slip conditions

The main goal of the research is to investigate the effects of temperature and concentration slip on the MHD Casson–Williamson nanofluid flow through a porous sheet, with a focus on viscous dissipation. This research has a novel approach, as it seeks to incorporate various factors that have not been previously studied in this context. Moreover, an inclined magnetic field from the outside is applied to the stretched surface. Besides this, it is considered that the viscosity of nanofluids depends on temperature while all other physical characteristics are taken to be constant. It is hypothesized that the expanding surface that stretched exponentially is what caused the flow motions of the nanofluid. There are several engineering applications for this type of study, which is based on MHD non-Newtonian nanofluid flow across a stretched surface with heat generation and viscous dissipation. They include solar energy storage, energy distribution, chemical reactors, and the manufacturing of polymers. To examine the heat transmission and mass transfer rates, tools like thermophoresis and Brownian motion were implemented. The flow problem is formally represented as a set of nonlinear PDEs that are then, through the use of dimensionless variables, converted into a set of ODEs. To numerically obtain the solution to the problem, the shifted Chebyshev polynomials of the third-kind approximation along with the spectral collocation technique are utilized. This process transforms the existing model into an algebraic equation system that was created as a restricted optimization problem, which is then optimized to obtain the solution and the unknown coefficients. For a variety of pertinent parameter values, the graphic and tabular representations of the concentration, temperature, velocities, rate of heat mass transfer, and shear stresses of nano-fluids at the surface of the sheet are shown. Finally, there is a remarkable amount of agreement between the earlier investigation and the comparison research that was conducted. The findings suggest that higher values of the magnetic parameter, Casson parameter, viscosity parameter, and suction parameter lead to a decrease in both velocity and boundary layer thickness. Furthermore, an increase in the viscosity parameter, Casson parameter, and magnetic parameter results in elevated values for both temperature and concentration.

Convective heating and mass transfer in Buongiorno model of nanofluid using spectral collocation method of shifted Chebyshev polynomial

In this article, the investigation is made to shed a light on the influence of convective boundary conditions on the boundary layer flow over a linearly stretching flat surface. The equations (partial differential equations) describing the model are transformed into system of nonlinear ordinary differential equations using similarity transformations. Equations contain various non-dimensional flow characterizing numbers viz. Prandtl number Pr, Lewis number Le, Biot number Bi, Brownian motion parameter Nb and thermophoresis parameter Nt. The influence of these parameters on thermal boundary layer, concentration distribution and temperature are analyzed in detail, by solving the equations using novel Shifted Chebyshev collocation method. The computed results, reduced Nusselt number, reduced Sherwood number, surface temperature and concentration profiles as functions of dimensionless numbers are validated by comparing the predicted results with available earlier findings (using other methods). To assert the convergence and stability of the scheme used (for much larger, but moderate, parameters values), predicted results are presented in various tabular forms. For presenting finer details of the computed values some results are also given graphically. The innovative semi-numerical scheme is robust and efficient compared with other conventional methods, used in previous studies and enables the analysis of the complex problem adequately.

Thermal damage analysis in tissue caused by electromagnetic radiation using space–time collocation method

Over the past half-century, the usage of external heat sources in medical applications has increased substantially. Controlling heat damage is essential for ensuring the efficacy of the treatment. Living tissues are highly non-homogeneous; hence, it is important to take into account the effects of local non-equilibrium on their thermal behavior. In the present study, two- and three- space dimensional time-space fractional single-phase-lag (SPL) and dual-phase-lag (DPL) models for bio-heat transfer in tissue are considered to study the thermal damage and temperature in tissue caused by electromagnetic radiation as an external heat source. The considered mathematical models are more general and consider non-Fourier as well as non-local effects. We obtain the numerical solution for the models by combining Gaussian RBFs and shifted Chebyshev polynomials in the space and time directions, respectively. The RBFs depend on Euclidean distance, so they can easily be used in multidimensional space domain, and the use of Chebyshev polynomials gives spectral accuracy in time direction. It is also explored how different parameters, such as blood perfusion rate Wb, phase lags τq, τt, and fractional derivatives α, β, affect the temperature distribution and thermal damage in the tissue.

Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

Enhanced shifted Tchebyshev operational matrix of derivatives: two spectral algorithms for solving even-order BVPs

Herein, new orthogonal polynomials have been generated from shifted Chebyshev polynomials that fulfill a given set of homogeneous boundary conditions and the necessary formulae have been established. Moreover, an integer order derivative operational matrix has been introduced. Then, the presented novel polynomials are used together with the two spectral methods, namely, the Galerkin and Tau methods, as the basis functions. The convergence and error analyses were introduced and proved. Finally, some even-order boundary value problems (BVPs) have been approximated using the presented method.

Numerical Investigation of Fredholm Fractional Integro-differential Equations by Least Squares Method and Compact Combination of Shifted Chebyshev Polynomials

In this study, linear Fredholm fractional integro-differential equations (FIDEs) are numerically solved, where the fractional derivative is considered in the Caputo sense. In this work, the least squares method (LSM) using a compact combination of shifted Chebyshev polynomials (SCP) of the first Kind is applied to solving a class of FIDEs. Our aim is to write the unknown function as a series of a linear combination of SCP, and then to reduce the problem to a system of linear algebraic equations, which will be solved for the unknown constants associated with the approximate solution, using MATLAB R2020a. Finally, numerical examples are presented to confirm the reliability, applicability, and efficiency of this method, in addition, various comparisons are also shown.

Shifted Chebyshev operational matrices to solve the fractional time-delay diffusion equation

In this paper, Chebyshev operational matrices collocation technique is proposed for solution of variable order derivative within the fractional time-delay diffusion equation. The beginning of this approach is based on the construction of the solution using the shifted Chebyshev polynomials with unknown coefficients. After that, we performed the Newton–Cotes nodal points, the Chebyshev polynomials operational matrices, and the collocation method for calculating the unknown coefficients. According to the described technique, we get an algebraic system of nonlinear equations which can be solved easily by using Newton’s iterative method. The efficiency and applicability of suggested approach are illustrated by some tested examples.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.

Fictitious finite integration method for solving high order partial differential equations

In this paper, we combine the finite integration method (FIM) with the technique of fictitious points to solve high order partial differential equations (HOPDEs). Two type of fictitious finite integration method (FFIM) are proposed and analyzed. The first type FFIM (FFIM-I) constructs the first order integration matrices by shifted Chebyshev polynomials so that the analytical high order integration matrices can be recursively obtained. Compared with the classical FIM, whose high order integration matrices are approximated directly through low order integration matrices, this FFIM-I leads to higher accuracy. Based on FFIM-I, the second type FFIM (FFIM-II) is proposed in which the fictitious points are located freely either inside or outside the problem domain. The FFIM-II can further improve the performance of FFIM-I in solving HOPDEs. The advantage of this fictitious finite integration method will be demonstrated by making comparisons among FIM, FFIM-I and FFIM-II through several 2D and 3D numerical experiments.

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

An efficient numerical approach for solving three-point Lane–Emden–Fowler boundary value problem

For three-point Lane–Emden–Fowler boundary value problems (LEFBVPs), we propose two robust algorithms consisting of Bernstein and shifted Chebyshev polynomials coupled with the collocation technique. The first algorithm uses the Bernstein collocation method with uniform collocation points, while the second is based on the shifted Chebyshev collocation method with roots as its collocation points. In both algorithms, the equivalent integral equations of three-point LEFBVPs are constructed. Then the approximation and collocation approach are used to generate a system of nonlinear equations. Then we implement Newton’s method to solve this system. The current method is different from the traditional method as we do not require to approximate u′ and u′′ appearing in the problem. It reduces not only the computational time but also the truncation error. The existence of a unique solution is discussed. Error analysis of both algorithms is demonstrated. A well known property of the proposed techniques is that it provides efficient solution for a few collocation points. The numerical results verify the same.