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Abstract In the current investigation, an error estimate has been proposed to solve the two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives based on the finite element/finite difference scheme. The time and space derivatives are based on the Riemann–Liouville and Riesz fractional derivatives, respectively. At first, the temporal variable has been discretized by a second-order difference scheme and then the space variable has been approximated by the finite element method (FEM). The analytical study shows that the presented scheme is unconditionally stable and convergent. Finally, some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of infinitely smooth and “nearly” locally supported radial basis functions (RBFs), derived from the general solution of a heat equation arising from the American option pricing problem. These basis functions are expressed in terms of “the repeated integrals of the complementary error function” and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model. We introduce an integral operator with a function-dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the RBF theory. We then show that using the introduced functions as expansion bases in the context of an RBF-based meshless collocation scheme, we could exactly impose the transparent boundary condition accompanying the heat equation. We prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular RBF families used in current literature. Some other properties of these bases such as their Fourier transforms as well as some useful representations in terms of positive Borel measures will also be discussed. [ABSTRACT FROM AUTHOR]

In this paper, meshless weak form techniques are applied to find the numerical solution of nonlinear biharmonic Sivashinsky equation arising in the alloy solidification problem. Stability and convergence analysis of time-discrete scheme are proved. An error analysis of meshless global weak form method based on radial point interpolation technique is proposed for this nonlinear biharmonic equation. In addition, a comparison between meshless global and local weak form methods is done from the perspective of accuracy and efficiency. The main purpose of this paper is to show that the meshless weak form techniques can be used for solving the nonlinear biharmonic partial differential equations especially Sivashinsky equation. The numerical results confirm the good efficiency of the proposed methods for solving this nonlinear biharmonic model. [ABSTRACT FROM AUTHOR]

This paper gives an error analysis for the method of lines (MOL) using generalized interpolating moving least squares (GIMLS) approximation. In this study, error bound for the time-dependent linear and nonlinear second-order differential equations in d -dimension will be obtained, when the GIMLS method is used for approximating the spatial variables. Also, the well-known Courant–Friedrichs–Lewy (CFL) condition will be derived in both cases (linear and nonlinear equations). Finally, numerical examples will be reported to confirm the ability of the proposed technique. [ABSTRACT FROM AUTHOR]

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function F δ 3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ 3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than F δ 3 / 2 . Note that there is a smart relation between δ 3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c 0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln ( c 0 r ) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to approximate the price of an American call option written on a dividend-paying stock close to expiry using an asymptotic analytic approach. We use the heat equation equivalent of the Black–Scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems. We extend the idea presented in [H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM Journal on Numerical Analysis 41 (6) (2003) 2081-2095.] in which a weakly singular memory-type transparent boundary condition (TBC) is obtained for the special case that the initial condition is equal to zero. We first derive this TBC in the general case and then focus on the outer problem in conjunction with an equivalent non-singular version of the TBC (dubbed ETBC) which is more tractable for analytical purposes. We then obtain the general solution of the outer problem in series form based on “the repeated integrals of the complementary error function” which also satisfies the introduced ETBC. As the next step, using the machinery of Poincaré asymptotic expansion and taking “time-to-expiry” as the expansion parameter, we find the general term of this series in closed form when the risk-free interest rate ( r ) is less than the dividend yield ( δ ). We also obtain the first five terms in the opposite case ( r > δ ) in a systematic manner. We also prove the convergence properties of the obtained series rigorously under some general conditions. Our numerical experiments based on the obtained asymptotic series, demonstrate the applicability and effectiveness of the results in valuation of a wide range of American option problems. [ABSTRACT FROM AUTHOR]

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L 2 norm which did not present in Mu and Zhang (2019) because of the artificial H 3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding. [ABSTRACT FROM AUTHOR]

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O ( τ 3 − α ) , 1 < α < 2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L ∞ -convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O ( τ ) . Since the shape functions of moving least squares approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed. An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction–subdiffusion equation. Firstly, we obtain a time discrete scheme based on a finite difference scheme, then we use the meshless Galerkin method, to approximate the spatial derivatives and obtain a full discrete scheme. In the proposed scheme, some integrals appear over the boundary and the domain of problem which will be approximated using Gauss–Legendre quadrature rule. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method. We show convergence order of the time discrete scheme is O ( τ γ ) . The aim of this paper is to obtain an error estimate and to show convergence for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In many studies in applied sciences and engineering one should find outer pseudo-inverse of a matrix. In this paper, we propose a new efficient algorithm for computing the outer pseudo-inverse of a matrix. We study the convergence analysis of the new algorithm. Finally, test problems and simulation results support the theoretical approach. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier–Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H 1 and L 2 norms, pressure in the L 2 norm, temperature in the discrete H 1 and L 2 norms, and the normal derivative of temperature in H − 1 / 2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the meshless local Petrov–Galerkin (MLPG) method is employed to solve the 2-D time-dependent Maxwell equations. The MLPG method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces. In the current work, the moving least square reproducing kernel (MLSRK) scheme is chosen to be the trial function. The method is applied for the unsteady Maxwell equations in different media. In the local weak form, by employing the difference operator for evolution in time and simultaneously in time and space, the semi-discrete and fully discrete schemes are obtained respectively. The error estimation is discussed for both the semi-discrete and fully-discrete numerical schemes for modelling the time-dependent Maxwell equations. We show that provided that the time step size is sufficiently small, the proposed scheme yields an error of in the norm for the square of error. The new scheme is implemented and the numerical results are provided to justify our theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, using the collocation method we solve the nonlinear fractional integro-differential equations (NFIDE) of the form: We study the convergence and the stability analysis of this method for . Some numerical examples are given to show the efficiency of the presented method. [Copyright &y& Elsevier]

In this paper, we propose a new approximation algorithm for solving generalized Lyapunov matrix equations. We also present a convergence analysis for this algorithm. In each step of this algorithm two standard Lyapunov matrix equations with real coefficient matrices should be solved. Then we determine the optimal parameter to minimize the corresponding spectral radius of iteration matrix to obtain fastest speed of convergence. Finally some numerical examples are given to prove the capability of the present algorithm and a comparison is made with the existing results. [ABSTRACT FROM AUTHOR]

Abstract: A generalization of moving least square reproducing kernel method is presented in this work. The moving least square reproducing kernel method is obtained by using a moving least square scheme but not in the discrete version. The resulted scheme provides a continuous basis which is able to reproduce any - order polynomial, and prepares a scheme that can approximate smooth functions with an optimal accuracy. On the other hand, considering the power of moving least square scheme in meshless approximation for the numerical solution of partial differential equations, the generalized moving least square approximation is able to approximate just in terms of node values where is an arbitrary linear operator. In this paper, a generalization of moving least square reproducing kernel method is presented which employs the generalized version of moving least square method. The method approximates a test functional, based on the values of nodes. The convergence rate of the method is measured in terms of dilation parameter of window function. The method is simpler and faster to implement than the classical ones where it does not use the shape function. Numerical tests are presented to confirm the theoretical results. The numerical results establish the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper a class of Birkhoff-type interpolation problems on arbitrary nodal points is studied. The explicit representation (characterization), the uniqueness and the error function are explicitly given. Furthermore, we apply the obtained Birkhoff-type interpolation method to find: (i) the numerical solution of high order initial-value problems (IVPs) and the corresponding error of this approximation, (ii) the approximation of some special functions with their explicit error functions, and (iii) new interpolatory type quadrature formulae of precision degree at least and . Numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results. The results reveal that the new method is effective, simple and accurate. [Copyright &y& Elsevier]

Abstract: A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

Abstract: This article investigates the numerical solution of the nonlinear integro-differential equations. The numerical scheme developed in the current paper is based on the moving least square method. The moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. It consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its closet points. Hence the method is a meshless method and does not need any background mesh or cell structures. The error analysis of the proposed method is provided. The validity and efficiency of the new method are demonstrated through several tests. [Copyright &y& Elsevier]

Abstract: Many problems in the areas of scientific computing and engineering applications can lead to the solution of the linear complementarity problem LCP . It is well known that the matrix multisplitting methods have been found very useful for solving LCP . In this article, by applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelaxation (SSOR) techniques, we introduce two class of synchronous matrix multisplitting methods to solve LCP . Convergence results for these two methods are presented when is an -matrix (and also an -matrix). Also the monotone convergence of the new methods is established. Finally, the numerical results show that the introduced methods are effective for solving the large and sparse linear complementary problems. [Copyright &y& Elsevier]

Abstract: In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. [Copyright &y& Elsevier]

Abstract: This note tries to study different solution sets of the interval linear matrix equation , where is a known square interval matrix of dimension , is a rectangular interval matrix of dimension , while the unknown matrix is also of dimension . Firstly, we show that Shary’s results for interval linear systems with a single right-hand side vector cannot be simply generalized to the case of interval linear systems of the form . Secondly, we give some analytical characterizations of the AE-solution sets of this interval matrix equation. We use a linear programming method in order to find the interval hull matrix. On the other hand, we propose the use of an interval Gaussian elimination to find an enclosure for the united solution set of this matrix equation, since the LU decomposition of is needed only once. Numerical examples have also been given. [Copyright &y& Elsevier]

Abstract: A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduced to a set of algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use. [ABSTRACT FROM AUTHOR]

Abstract: During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov–Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains. [Copyright &y& Elsevier]

Abstract: A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function and the unknown coefficient in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method. [Copyright &y& Elsevier]

Abstract: Computing a function of an -by- matrix is a frequently occurring problem in control theory and other applications. In this paper we introduce an effective approach for the determination of matrix function . We propose a new technique which is based on the extension of Newton divided difference and the interpolation technique of Hermite and using the eigenvalues of the given matrix . The new algorithm is tested on several problems to show the efficiency of the presented method. Finally, the application of this method in control theory is highlighted. [Copyright &y& Elsevier]

Abstract: The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. [Copyright &y& Elsevier]

Abstract: In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of Cauchy reaction–diffusion problem is presented by means of variational iteration method. Reaction–diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields. Application of variational iteration technique to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computations. [Copyright &y& Elsevier]

Abstract: In this work we will consider He''s variational iteration method for solving second-order initial value problems. We will discuss the use of this approach for solving several important partial differential equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This procedure is a powerful tool for solving the large amount of problems. Using the variational iteration method, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a sequence of functions which converges to the exact solution of the problem. Our emphasis will be on the convergence of the variational iteration method. In the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models. [Copyright &y& Elsevier]

Hereditary effects are commonly observed in diverse scientific domains such as engineering, economics, biology, mathematics, and physics. In the model of atomic irradiation of solids with unbounded cross-sectional areas, determining the average number of atoms displaced has been achieved through delay systems that incorporate the consideration of past states. In this study, we employ the discrete collocation method using local radial basis functions to numerically solve Abel-type integral equations with a delay argument. This method balances accuracy, efficiency, and flexibility and is well-suited for complex practical problems as it requires less memory and computational volume compared to its global counterpart. Instead of approximating the solution at all points of the domain, the method considers a set of nodes in the neighborhood of a certain point. As a result, the method can be readily implemented on a standard personal computer without requiring high-end specifications. To calculate the singular integrals the nonuniform composite Gauss–Legendre numerical integration rule is employed. We discuss the error analysis and convergence rate of the offered scheme and test it with several numerical examples. The results obtained are also consistent with the theoretical error analysis expectations. [ABSTRACT FROM AUTHOR]

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains. [ABSTRACT FROM AUTHOR]

The fractional PDEs based upon the distributed-order fractional derivative have several applications in physics. The two-dimensional time-space distributed-order weakly singular integro-partial differential model is investigated by a combination of finite difference and Galerkin spectral methods. A second-order finite difference formula is employed to approximate the temporal variable. In this stage, the stability and convergence of the semi-discrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space variable. Also, in this step, the error estimate of the full-discrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We develop a meshless numerical procedure to simulate the groundwater equation (GWE). The used technique is based on the interpolating element free Galerkin (IEFG) method. The interpolating moving least squares (IMLS) approximation produces a set of functions such that they are well-known as " shape functions ". The IEFG technique employs the shape functions of IMLS approximation. The shape functions of IMLS approximation vanish on the boundary and also they satisfy the property of the Kronecker Delta function. Thus, Dirichlet boundary conditions can be exactly imposed. In this paper, we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method. The numerical results confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a computational method for solving Fredholm integral equations of the second kind with logarithmic kernels. The method is based on the discrete Galerkin method with the shape functions of the moving least squares (MLS) approximation constructed on scattered points as basis. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule for approximating logarithm-like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations. The proposed method is meshless, since it does not require any background mesh or domain elements. The error analysis of the method is provided. The scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of Laplace’s equation with linear Robin boundary conditions. Finally, numerical examples are included to show the validity and efficiency of the new technique. [Copyright &y& Elsevier]

Abstract: A numerical direct method for solving a general class of fractional optimal control problems (FOCPs) is presented. In the discussed FOCP, the fractional derivative in the dynamical system is considered in the Caputo sense. To solve the problem, first the FOCP is transformed into an equivalent variational problem, then using the Legendre orthonormal basis, the problem is reduced to the problem of solving a system of algebraic equations. With the aid of an operational matrix of Riemann–Liouville fractional integration, Gauss quadrature formula and Newton’s iterative method for solving a system of algebraic equations, the problem is solved approximately. Approximations achieved by this method satisfy all the initial conditions of the problem which is an important property. The convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique. [Copyright &y& Elsevier]

Abstract: In this paper, we present a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on a non-rectangular domain. The method utilizes radial basis functions (RBFs) constructed on scattered points as a basis in the discrete collocation method. The proposed scheme is meshless, since it does not need any domain element and so it is independent of the geometry of the domain. The method reduces the solution of the two-dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations. Error analysis is presented for this method. Finally, numerical examples are included to show the validity and efficiency of the new technique. [Copyright &y& Elsevier]

Abstract: Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation. [Copyright &y& Elsevier]