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Investigation of spectral (collocation or Galerkin) methods for the solution approximation of different classes of optimal control problems have had been increased in recent years. In this research work, we proposed and analyzed an applicable Legendre spectral collocation approach, based on an efficient space of fractional basis functions, to compute numerically the solution of fractional optimal control problems. We first introduce a fractional Lagrange interpolation and then approximate the unknown control and state by this new set of fractional interpolation functions. The problem is discretized in terms of shifted Legendre–Gauss collocation points and two approaches are suggested to calculate the exact fractional differentiation matrix. Also, we prove that the approximate solutions of the obtained discretized problem converge to the optimal solution of the main problem by assuming some mild conditions. The method is implemented for three numerical test problems to show the claimed efficiency and capability.

This paper contributes to develop a highly accurate numerical method for solving two-dimensional mass transfer equations during convective air drying of apple slices. The numerical scheme is based on the interpolating the solution of the mentioned equations over the roots of the orthogonal Jacobi polynomials (i.e., the Jacobi-Gauss-Lobatto points) in a nodal form. Moreover, for speeding up the procedure of numerical technique, operational matrices of differentiation are applied to discretize the derivatives of both spatial and temporal variables. After implementing the proposed technique, two-dimensional mass transfer equations would be transferred into the associated systems of linear algebraic equations which can be solved by appropriate iterative solvers such as robust Krylov subspace iterative methods. Some constructed artificial examples are provided to show the efficiency and applicability of the Jacobi pseudo-spectral method for solving two-dimensional mass transfer equations. Finally, a real example is considered and numerical results are validated by the experimental data which confirm the accuracy of the presented numerical approach.

Increased attention has been paid on numerical modeling of interface problems as its wide applications in various aspects of science. Motivated by enhancing the application of the reproducing kernel, we focus on establishing a broken cubic spline spaces and developing a fourth-order numerical scheme for one-dimensional elliptic interface problems in this paper. The method is based on the least-squares method and broken cubic spline spaces. A new set of basis of the established broken cubic spline space are obtained by integrating the reproducing kernel function of W 2 1 . We prove that the proposed method is stable. The optimal convergence orders under H 2 , H 1 and L 2 norms are also discussed. Our main contribution is that our symmetric method is stable and can be naturally extended to higher order schemes. Finally, our theoretical findings are verified by several numerical experiments. In addition, some comparisons of the proposed method with difference potentials methods, reproducing kernel methods and immersed finite element methods are given.

We study a higher-order Legendre reproducing kernel method (LRKM) for singular two-point boundary value problems (BVPs). Our method relies upon Legendre polynomials. We carry out error estimatation by using interpolation theory. Various numerical tests including linear and nonlinear problems are performed to reveal the stability and efficiency of the new scheme. Therewith, as an application of the proposed approach, a famous time-dependent partial differential equation, namely Allen–Cahn test problem, is transfomed into the associated system of linear BVPs by localizing in temporal variable and is challenged numerically in details by the LRKM.

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments.

Purpose The purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations. Design/methodology/approach In this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations. Findings The proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries. Originality/value This study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

In this paper, a numerical method is applied for solving delay fractional optimal control problems (DFOCPs). The fractional derivative is described in the Caputo sense. Since the fractional derivative of Muntz polynomials can be expressed in terms of the same polynomials, those polynomials can accurately represent properties of fractional calculus. In some situations such as in the frequency response based analysis of control systems containing a time-delay, it is necessary to substitute exponential function with an approximation in the form of a rational function. The most common approximation is the Pade approximation. At the first step, using Pade approximation, the delay problem is transformed to a non-delay problem. Next, using the operational matrix of the fractional derivative of Muntz polynomials and pseudospectral method, fractional optimal control problem (FOCP) is reduced to a nonlinear programming problem. Some numerical examples are given to illustrate the effectiveness of the proposed scheme.

In the present paper we use the generalized Euler-Maclaurin summation formula to study the convergence and to solve weakly singular Fredholm and Volterra integral equations. Since these equations have different nature, the proposed convergence analysis for each equation has a different structure. Moreover, as an application of this summation formula, we consider the numerical solution of the fractional ordinary differential equations (FODEs) by transforming FODEs into the associated weakly singular Volterra integral equations of the first kind. Some numerical illustrations are designed to depict the accuracy and versatility of the idea.

This paper contributes to investigate the Jacobi spectral and pseudo-spectral Galerkin techniques to solve a general form of nonlinear weakly singular Volterra integro-differential equations of the first order. By applying some suitable change of variables, we have made the solution of the mentioned equations to be smooth. Then, by applying the spectral and pseudo-spectral Jacobi Galerkin schemes, accurate solutions are computed efficiently. Rigorous convergence analysis associated with both the spectral and pseudo-spectral Jacobi Galerkin approaches are discussed in detail. Some numerical test problems are given to depict the accuracy of the presented numerical schemes with respect to some recent approximate methods in the literature.

This paper is devoted to solve a class of weakly singular Volterra integro-differential equations with noncompact kernels. Since the solution of such these models may has singularities, so the classical spectral methods lose their high accuracy for solving them. Innovation of this article is that, we use the generalized mapped nodal Laguerre spectral collocation method to deal with the singularity. Therefore, we can make good use of the advantages of the mapped Laguerre functions. The best advantage of the proposed method is its robustness for solving problems that have singularity near the left (or right) boundary of the computational interval. We present the construction and analysis of the generalized log orthogonal Laguerre functions collocation method in this paper and some numerical examples with nonsmooth solutions are included to show the efficiency of the suggested numerical scheme with respect to the classical Jacobi spectral methods.

Based on the least square method, we proposed a new algorithm to obtain the solution of the second kind regular and weakly singular Volterra-Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in details. Numerical experiments verify the theoretical findings. Meanwhile this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non-smooth solutions are also considered and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR.

The present paper deals with a variant of the generalized Euler–Maclaurin summation formula for the product integration rule. The main idea lies on introducing a mesh associated with the integral of the square root of the weight function. We construct the set of nodes and implement it for approximating the considered integrals. It is shown theoretically that the proposed quadrature rule is of fourth order. The results of the provided numerical examples confirm the theoretical prediction. Moreover, applications of the suggested scheme for approximating some real test problems such as weakly singular integrals, including a particular case of the Fermi–Dirac integral, are investigated.

The present research is contemplated proposing a numerical solution of multi-dimensional hyperbolic telegraph equations with appropriate initial time and boundary space conditions. The truncated Hermite series with unknown coefficients are used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing the considered one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) telegraph equations is based on the collocation method together with the Hermite operational matrices of derivatives. The resulted systems of linear algebraic equations are solved by some efficient methods such as LU factorization. The solution of the algebraic system contains the coefficients of the truncated Hermite series. Numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme. Comparisons of numerical results associated to the proposed method with some of the existing numerical methods confirm that the method is accurate and fast experimentally.

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.

In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

This paper is devoted to develop a new matrix scheme for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions. We first transform these equations into equivalent integro partial differential equations (PDEs). Such these integro-PDEs contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner. Subsequently, all the existing known and unknown functions in the latter equations are approximated by Bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro-PDEs into the associated algebraic generalized Sylvester equations. For solving these algebraic equations, an efficient Krylov subspace iterative method (i.e., BICGSTAB) is implemented. Two numerical examples are given to demonstrate the efficiency, accuracy, and versatility of the proposed method.

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev&ndash, Gauss&ndash, Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

In this paper, we developed a new and efficient numerical method for solving advection–diffusion equations. The method is based on the integration of the considered advection–diffusion equation. By integration, we transform the advection diffusion equation into the equivalent integral equation which the integral equation contains initial and boundary conditions. Afterward, the integral equation would be transformed into the system of linear algebraic equations by using a hybrid of Chebyshev and Block-pulse functions and their operational matrix of integration. The system of linear algebraic equations can be solved by direct or iterative methods. We presented some theorems to show the error approximation of the hybrid function. Three numerical examples are considered to investigate the applicability and simplicity of the method. The numerical results confirm that the method is fast, stable, and exponentially accurate.

This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one- and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub–Kahan and reducing the linear operator $${\mathcal {L}}$$ to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

This present investigation is contemplated to provide Legendre spectral collocation method for solving multi-Pantograph delay boundary value problems (BVPs). In this regard, an equivalent integral form of such BVPs has been considered. The proposed method is based on Legendre–Gauss collocation nodes and Legendre–Gauss quadrature rule. Convergence analysis associated to the presented scheme has been provided to show its applicability theoretically. Some numerical examples are given to demonstrate the efficiency, accuracy, and versatility of our method. Numerical results confirm the theoretical predictions and are superior with respect to several recent numerical methods including Hermite collocation approach, Laguerre collocation technique and the reproducing kernel method.

Spectral and pseudo-spectral Galerkin techniques, by using the standard Jacobi polynomials, are implemented to calculate numerically the solutions of pantograph type Volterra delay integro-differential equations that have kernels with the property of weak singularity. Because of the complex structure of the considered problems, pseudo-spectral Galerkin approaches are more desirable with respect to the spectral Galerkin approaches, since they have the property of integral approximator by using high order convergent Gauss quadrature formulas. A deep and detailed analysis of convergence of the numerical solutions to the exact solutions are given under some mild conditions such as smoothness of the solutions. Some test problems are illustrated and efficiency of the suggested numerical approach is investigated with respect to a recently proposed Jacobi pseudo-spectral collocation technique via some figures and tables experimentally.

In this paper, the space-time Jacobi spectral collocation method (JSC Method) is used to solve the time-fractional nonlinear Schr o ¨ dinger equations subject to the appropriate initial and boundary conditions. At first, the considered problem is transformed into the associated system of nonlinear Volterra integro partial differential equations (PDEs) with weakly singular kernels by the definition and related properties of fractional derivative and integral operators. Therefore, by collocating the associated system of integro-PDEs in both of the space and time variables together with approximating the existing integral in the equation using the Jacobi-Gauss-Type quadrature formula, then the problem is reduced to a set of nonlinear algebraic equations. We can consider solving the system by some robust iterative solvers. In order to support the convergence of the proposed method, we provided some numerical examples and calculated their L∞ norm and weighted L2 norm at the end of the article.

This article provides an efficient matrix approach by using Euler approximation for solving numerically the two-dimensional diffusion and telegraph equations subject to the Dirichlet boundary conditions. First, the main equation is reduced into partial integro-differential equations (PIDEs) and then operational matrices of differentiation and integration of Euler polynomials transform those PIDEs into algebraic generalized Sylvester equations. The inclusion of several test examples confirms the predicted accuracy and effectiveness of the method. Comparison of obtained numerical results is made with some earlier works (Zogheib and Tohidi, 2016, Singh et al., 2018).

A novel local meshfree method is proposed for simulating cash-or-nothing and asset-or-nothing options, at which the initial condition is discontinuous. The novelty of the scheme is in the use of Hermite radial basis function (RBF) interpolation on set of local nodes. We also construct the RBF generated Hermite finite difference (HFD) formulas and subsequently obtain their associated differentiation matrices (DMs). The developed accuracy equipped with a θ -shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests.

This article deals with the implementation of Jacobi spectral and pseudospectral Galerkin approaches for solving weakly singular Volterra integral equations with noncompact kernels. The computational domain of the main problem and intervals of the involved integrals are transformed into [-1,1] by some appropriate change of variables. Therefore, one can apply the Jacobi Galerkin method and impose the inner product conditions on the residual of the considered equation. Also, Jacobi Gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the Jacobi weight function in the pseudospectral Galerkin method. For both cases of spectral and pseudospectral Galerkin schemes, rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in L ∞ and weighted L 2 norms. Several numerical examples are also given to confirm the theoretical prediction.

High-order methods have attracted considerable attention in the CFD fields because of their potential in achieving higher accuracy with a lower cost than low order methods. In this paper, a new class of high order spectral volume method based on Chebyshev polynomials as an approximation function called Chebyshev Spectral Volume (CSV) method is presented for hyperbolic conservation laws in quadrilateral meshes. As we know, the hyperbolic conservation law may contain discontinuities even if the initial conditions are smooth. To capture these discontinuities, we propose a new limiter to reconstruct the numerical approximation on the target cell, which is marked as a troubled cell. It is based on the Chebyshev polynomials in the target and its immediate neighboring cells to achieve a high order of accuracy and non-oscillatory properties. Also, a new indicator is introduced to detect troubled cells. This indicator depends neither on the numerical order of accuracy nor on a problem-dependent parameter. A series of 1D and 2D scalar and system conservation laws are presented to evaluate the performance of the CSV method coupled with the new limiter and indicator.