جستجوی مقالات مرتبط با کلیدواژه « Collocation method » در نشریات گروه « ریاضی »

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.

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.

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.

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.

In this paper, we present a well-organized strategy to estimate the fractional advection-diffusion equations, which is an important class of equations that arises in many application fields. Thus,  Lagrange square interpolation is applied in the discretization of the fractional temporal derivative, and the weighted and shifted Legendre polynomials as operators are exploited to discretize the spatial fractional derivatives of the space-fractional term in multi-termtime fractional advection-diffusion model. The privilege of the numerical method is the orthogonality of Legendre polynomials and its operational matrices which reduces time computation and increases speed. A second-order implicit technique is given, and its stability and convergence are investigated. Finally, we propose three numerical examples to check the validity and numerical results    to illustrate the precision and efficiency of the new approach.

In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equations using Caputo sense. We obtain the integral form of the problem and transform it into a system of linear alge-braic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method’s uniqueness as well as the convergence of the method. Numerical examples show that the developed method is efficient in problem-solving and competes favorably with the existing method.

This paper aims to present a new and efficient numerical method to approximate the solution of the fractional model of human T-cell lymphotropic virus I (HTLV-I) infection $CD4^+T$-cells. The approximate solution of the model is obtained using the shifted Chebyshev collocation spectral method. This model relates to the class of nonlinear ordinary differential equations. The proposed algorithm reduces the Caputo sense fractional model to a system of nonlinear algebraic equations that can be solved numerically. The convergence of the proposed method is investigated. The graphical result is compared with existing numerical methods reported in the literature to indicate the efficiency and reliability of the presented method.

In this paper, we use the CAS wavelets as basis functions to numerically solve a system of nonlinear Fredholm integro-differential equations. To simplify the problem, we transform the system into a system of algebraic equations using the collocation method and operational matrices. We show the convergence of the presented method and then demonstrate its high accuracy with several illustrative examples. This approach is particularly effective for equations that admit periodic functions because the employed basis CAS functions are inherently periodic. Throughout our numerical examples, we observe that this method provides exact solutions for equations with trigonometric functions at a lower computational cost when compared to other methods.

The presented paper examines a numerical method for solving Lane-Emden type equations based on Flatlet oblique multiwavelet properties. In this paper, using the Flatlet multiwavelet features, an operator matrix is created and then the Lane-Emden equation reduces to a set of algebraic equations. Also, comparing the results presented in previous articles, it is observed that this wavelet due to having different high ranks, has the ability to solve this problem more accurately than other methods.

This paper discusses a numerical method for solving a first-kind Volterra integral equations system. Because of the ill-posedness of these equations, we need to apply an efficient computational method to discrete them to the system of algebraic equations. An expansion method known as the Chebyshev collocation method, based on the Chebyshev polynomials of the third kind, is employed to convert the system of integral equations to the linear algebraic system of equations. By solving the algebraic system, we conclude an approximate solution. Some numerical results support the accuracy and efficiency of the stated method.

‎In this paper‎, ‎an efficient and robust approach based on the Chebyshev collocation method and Teaching-Learning-Based Optimization (TLBO) is utilized to solve the Optimal Control Problem (OCP) of reservoir sedimentation on Golestan dam in Gonbad Kavous City‎, ‎Iran‎. ‎The discretized method employs Mth degree of Lagrange polynomial approximation for an unknown variable and Gauss-Legendre integration‎. ‎The OCP yields a nonlinear programming problem (NLP)‎, ‎and then this NLP is solved by TLBO‎. ‎Numerical implementations are given to demonstrate this approach yields more acceptable and the accurate results‎. ‎Furthermore‎, ‎it is found that filling the dam with sediment decreases the water storage‎, ‎increases dam maintenance costs‎, ‎and also decreases the stability of the dam over a period of 40 years‎. ‎Our results show that the Golestan dam will gain development with the construction of the new reservoir‎.

In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on Chebyshev cardinal polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the Chebyshev cardinal functions are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

In this paper, a spectral collocation method for solving nonlinear pantograph type delay differential equations is presented. The basis functions used for the spectral analysis are based on Chebyshev, Legendre, and Jacobi polynomials. By using the collocation points and operations matrices of required functions such as derivative functions and delays of unknown functions, the method transforms the problem into a system of nonlinear algebraic equations. The solutions of this nonlinear system determine the coefficients of the assumed solution. The method is explained by numerical examples and the results are compared with the available methods in the literature. It is seen from the applications that our method gives more efficient results than that of the reported methods.