collocation method

In this paper, we introduce an effective multistep collocation method for solving delay differential equations (DDEs) with constant delays. We determine the convergence properties of the proposed method for delay differential equations with solutions in appropriate Sobolev spaces and show that the proposed scheme enjoys spectral accuracy. Numerical results show that the proposed method can be implemented efficiently and accurately for various DDE model problems.

‎This work applies rational Gegenbauer functions and a collocation scheme to solve the governing equation for two-dimensional fluid flow near a stagnation point‎, ‎known as Hiemenz flow‎. ‎We utilize a truncated series expansion of rational Gegenbauer functions on the semi-infinite interval and Gegenbauer–Gauss points to reduce the problem to a set of nonlinear algebraic equations‎. ‎Newton's iteration technique is employed to solve these algebraic equations‎. ‎The scheme is straightforward to implement‎, ‎and our new results are compared with established numerical results‎, ‎demonstrating the method's effectiveness and accuracy‎.

A numerical solution for the second kind singular integral equations with Cauchy kernel is developed using the collocation method. To achieve this, we approximate the Cauchy integral equation using the collocation method and Legendre orthogonal polynomial expansions. The accuracy of our proposed method is assessed through convergence and error analysis. Finally, several numerical examples are presented to  demonstrate the high efficiency of the method.

Water pollution can have many adverse effects on the environment and human health. The study of the transmission of water pollutants over a finite lifespan is carried out using an optimal control problem (OCP), with the system governed by ordinary differential equations. By utilizing the collocation approach, the OCP is transmuted to a nonlinear programming problem, and then the mountain Gazelle algorithm is applied to determine the optimal control and state solutions. A practical study demonstrates the effect of treatment on reducing water pollutants during a finite time.

In this paper, the functional Volterra integral equations of the Hammerstein type are studied. First, some conditions that ensure the existence and uniqueness of the solutions to these equations within the space of square-integrable functions are established and then the Euler operational matrix of integration is constructed and applied within the collocation method for approximating the solutions. This approach transforms the integral equation into a set of nonlinear algebraic equations, which can be efficiently solved by employing standard numerical methods like Newton's method or Picard iteration. One significant advantage of this method lies in its ability to avoid the need for direct integration to discretize the integral operator. Error estimates are provided and two illustrative examples are included to demonstrate the method’s effectiveness and practical applicability.

In this paper, we aim to introduce aweighted orthogonal system on the half-line based on theexponential Gegenbauer functions. We use these functions incollocation method to solve MHD Falkner-Skan equation, whicharises in the study of laminar boundary layers exhibitingsimilarity on the semi-infinite domain.This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming the domain of the problem to a finite domain. We make a comparisonbetween the results of the proposed system with the numericalresults to show that the present methodhas an acceptable accuracy.

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.

This study presents a method to solve weakly singular Volterra integral equations using an approximation approach. The method relies on Chelyshkov wavelet polynomials. The characteristics of the Chelyshkov wavelet are presented. By employing these polynomials, the singular Volterra integral equation is transformed into a set of algebraic equations that need to be solved. Subsequently, numerical analysis is introduced, followed by some examples and a comparison of them with existing methods to demonstrate the soundness and practicality of our method.

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 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.

The hyperbolic partial differential equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with variable coefficients. The Goursat PDE is transformed into a second kind of linear Volterra in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method’s effectiveness and precision.

‎In this paper‎, ‎a special system of non-linear Abel integral equations (SNAIEs) is studied which arises in astrophysics‎. ‎Here‎, ‎the well-known collocation method is extended to obtain approximate solutions of the SNAIEs‎. ‎The existence and uniqueness conditions of the solution are investigated‎. ‎Finally‎, ‎some examples are solved to illustrate the accuracy and efficiency of the proposed method.‎

A hybrid method utilizing the collocation technique with B-splines and Lie-Trotter splitting algorithm applied for 3 model problems which include a single solitary wave,  two solitary wave interaction, and a Maxwellian initial condition is designed for getting the approximate solutions for the generalized equal width (GEW) equation. Initially, the considered problem has been split into 2 sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the  terms of time.  After, numerical schemes have been constructed for these sub-equations utilizing the finite element method (FEM) together with quintic B-splines. Lie-Trotter splitting technique $\hat{A}o\hat{B}$ has been  used to generate approximate solutions of the main equation. The stability analysis of acquired numerical schemes has been examined by the Von Neumann method. Also, the error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$, $I_2$, and $I_3$, respectively are calculated to illustrate how perfect solutions this new algorithm applied to the problem generates and the ones produced are compared with those in the literature. These new results exhibit that the algorithm presented in this paper is more accurate and successful, and easily applicable to other non-linear partial differential equations (PDEs) as the present equation.

The presented paper investigates a new numerical method based on the characteristics of flatlet oblique multiwavelets for solving fractional Volterra integro-differential equations, in this method, first using the dual bases of the flatlet multiwavelets, the operator matrices are made for the derivative of fractional order and Volterra integral. Then, the fractional Volterra integro-differential equation reduces to a set of algebraic equations which can be easily solved. The error analysis and convergence of the presented method are discussed. Also, numerical examples will indicate the acceptable accuracy of the proposed method, which is compared with the methods used by other researchers.

Due to the importance of the generalized nonlinear Klein-Gordon equation (NL-KGE) in describing the behavior of light waves and nonlinear optical materials, including phenomena such as optical switching by manipulating the dispersion and nonlinearity of optical fibers and stable solitons,  we explain the approximation of this model by evaluating different classical and fractional terms  in this paper. To estimate the fundamental function, we use a first-order finite difference approach in the temporal direction and a collocation method based on Gegenbauer polynomials (GP) in the spatial direction to solve the NL-KGE model. Moreover, the stability and convergence analysis is proved by examining the order of the new method in the time direction as $\mathcal{O}( \delta t )$. To demonstrate the efficiency of this design, we presented numerical examples and made comparisons with other methods in the literature.

Haar-Sinc spectral method is used for the numerical approximation of time fractional Burgers’equations with variable and constant coefficients. The main idea in this method is using a linear discretization of time and space by combination of Haar and Sinc functions, respectively. While implementing the method, the operational matrices of the fractional integral of the fractional Haar functions are made, and by using them, an algebraic equation is obtained. Then, using the collocation method, the algebraic equation is converted into a system of equations, and after solving the system with Maple software, the numerical results of the problem is obtained. The accuracy and speed of the proposed algorithm are tested by obtaining L∞, L2 error and the convergence rate.

This work considers a numerical method based on the 2D-hybrid block-pulse functions and normalized Bernstein polynomials to solve 2D-nonlinear Fredholm integral equations of the second type. These problems are reduced to a system of nonlinear algebraic equations and solved by Newton's iterative method along with the numerical integration and collocation methods. Also, the convergence theorem for this algorithm is proved. Finally, some numerical examples are given to show the effectiveness and simplicity of the proposed method.

Reservoir sedimentation increases economic cost and overflow of dam water. An optimal control problem (OCP) with singularly perturbed equations of motion is perused in the fields of sediment management during a finite lifespan. Subsequently the OCP is transformed to a nonlinear programming problem by utilizing a collocation approach, and then we employed the imperialist competitive algorithm to improve the execution time and decision. So, the solutions of the optimal control and fast state as well as the maximization of net present value of dam operations are obtained. An illustrative practical study demonstrated that sedimentation management is economically favourable for volume of confined water and total amount in remaining storage and effectiveness of the propounded approach.