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The main aim of the current paper is to propose a new truly meshless numerical technique to solve the one- and two-dimensional elliptic interface problems. The employed meshless approach is based on a new class of MLS approximation. The numerical procedure is based on the interpolating stabilized MLS (ISMLS) approximation. The new shape functions that have been made by the ISMLS technique, have the δ -Kronecker property thus the Dirichlet boundary conditions can be applied, directly. In the current investigation, we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ISMLS technique. Several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure. Moreover, test problems show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]

The element free Galerkin technique is a meshless method based on the variational weak form in which the test and trial functions are the shape functions of moving least squares approximation. Since the shape functions of moving least squares approximation do not have the Kronecker property thus the Dirichlet boundary condition can not be applied directly and also in this case obtaining an error estimate is not simple. The main aim of the current paper is to propose an error estimate for the extracted numerical scheme from the element free Galerkin method. To this end, we select the fractional cable equation with Dirichlet boundary condition. Firstly, we obtain a time-discrete scheme based on a finite difference formula with convergence order O ( τ 1 + min ⁡ { α , β } ) , then we use the meshless element free Galerkin method, to discrete the space direction and obtain a full-discrete scheme. Also, for calculating the appeared integrals over the boundary and the domain of problem the Gauss–Legendre quadrature rule has been used. In the next, we change the main problem with Dirichlet boundary condition to a new problem with Robin boundary condition. Then, we show that the new technique is unconditionally stable and convergent using the energy method. We show convergence orders of the time discrete scheme and the full discrete scheme are O ( τ 1 + min ⁡ { α , β } ) and O ( r p + 1 + τ 1 + min ⁡ { α , β } ) , respectively. So, we can say that the main aim of this paper is as follows, (1) Transferring the main problem with Dirichlet boundary condition (old problem) to a problem with Robin boundary condition (new problem), (2) Showing that with special condition (when σ → + ∞ ) the solution of the new problem is convergent to the solution of the old problem, (3) Obtaining an error estimate for the new problem. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

Recently, Zayernouri and Karniadakis in (2013) [78] investigated two classes of fractional Sturm–Liouville eigenvalue problems on compact interval [ a , b ] in more detail. They were the first authors who not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions. Until now, to the best of our knowledge, fractional Sturm–Liouville eigenvalue problems on non-compact interval, such as [ 0 , + ∞ ) are not analyzed. So, our aim in this paper is to study these problems in detail. To do so, we study at first fractional Sturm–Liouville operators (FSLOs) of the confluent hypergeometric differential equations of the first kind and then two special cases of FSLOs: FSLOs-1 and FSLOs-2 are considered. After this, we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues. Also, we derive two fractional types of the associated Laguerre differential equations. Due to the non-polynomial nature of the eigenfunctions obtained from the two fractional associated Laguerre differential equations, they are defined as generalized associated Laguerre functions of the first and second kinds, GALFs-1 and GALFs-2. Furthermore, we prove that these fractional Sturm–Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to FSLOs-1 and FSLOs-2 and form two sets of non-polynomial bases. At the end, two new quadrature rules and L 2 -orthogonal projections with respect to and based on GALFs-1 and GALFs-2 are introduced. The upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples. Finally, some fractional differential equations are provided and analyzed numerically. [ABSTRACT FROM AUTHOR]

Three inverse problems of reconstructing the time-dependent, spacewise- dependent and both initial condition and spacewise-dependent heat source in the one-dimensional heat equation are considered. These problems are reformulated by eliminating the unknown functions using some special assumptions concerning the points in space or time as additional measurements. Then direct techniques are proposed to solve the non-classical boundary value problems. For obtaining the robust and stable approximations, Bernstein multi-scaling and B-spline basis functions in the context of the Ritz–Galerkin method are utilized to immediate passage from differential equations to algebraic equations and afterwards, a Newton-type method is used to produce the admissible solution. The numerical convergence and stability are discussed in the test examples to show that the presented schemes provide accurate and acceptable approximations. [ABSTRACT FROM AUTHOR]

Abstract: In the last three decades, Sinc numerical methods have been extensively used for solving differential equations, not only because of their exponential convergence rate, but also due to their desirable behavior toward problems with singularities. This paper illustrates the application of Sinc-collocation and Sinc-Galerkin methods to the approximate solution of the two-dimensional time dependent Schrödinger equation with nonhomogeneous boundary conditions. Some numerical examples are presented and the proposed methods are compared with each other. [Copyright &y& Elsevier]

Abstract: Interest in meshless methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in science and engineering. In this paper, we present the moving least square radial reproducing polynomial (MLSRRP) meshless method as a generalization of the moving least square reproducing kernel particle method (MLSRKPM). The proposed method is established upon the extension of the MLSRKPM basis by using the radial basis functions. Some important properties of the shape functions are discussed. An interpolation error estimate is given to assess the convergence rate of the approximation. Also, for some class of time-dependent partial differential equations, the error estimate is acquired. The efficiency of the present method is examined by several test problems. The studied method is applied to the parabolic two-dimensional transient heat conduction equation and the hyperbolic two-dimensional sine-Gordon equation which are discretized by the aid of the meshless local Petrov–Galerkin (MLPG) method. [Copyright &y& Elsevier]

Abstract: In this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems (FVPs) with completely free boundary conditions have an extremum. The fractional derivatives are defined in the Caputo sense. First we present the necessary conditions for the problem with only one dependent variable, and then we generalize them to problems with multiple dependent variables. We also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables. It is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives, the new results reduce to the known necessary conditions. Some examples are presented to demonstrate the applicability of the new formulations. [Copyright &y& Elsevier]

Abstract: Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use. [Copyright &y& Elsevier]

Third-order boundary value problems (BVPs) were discussed in many papers in recent years. The existence of solutions to these problems is shown under certain conditions. Finding the exact or approximate solution of these problems is not easy especially when the boundary conditions are non-linear. In this paper, we present a method for solving a class of third-order BVPs with non-linear boundary conditions. The decomposition procedure of Adomian is used for solving these problems and the advantages of this method over the other existing methods are shown. Numerical results are presented to show the efficiency of the developed method. [ABSTRACT FROM AUTHOR]

Abstract: A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods. [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, by using the extended Sturm–Liouville theorem for symmetric functions, we introduce the differential equation as a generalization of the differential equation of trigonometric sequences and for and obtain its explicit solution in a simple trigonometric form. We then prove that the obtained sequence of solutions is orthogonal with respect to the constant weight function on and compute its norm square value explicitly. One of the important advantages of this generalization is to find some new infinite series. A practical example is given in this sense. [Copyright &y& Elsevier]

Abstract: Hyperbolic partial differential equations with an integral condition serve as models in many branches of physics and technology. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional nonlocal hyperbolic equation is presented by the method of lines. The method of lines (MOL) is a general way of viewing a partial differential equation as a system of ordinary differential equations. The partial derivatives with respect to the space variables are discretized to obtain a system of ODEs in the time variable and then a proper initial value software can be used to solve this ODE system. We propose two forms of MOL for solving the described problem. Several numerical examples and also some comparisons with finite difference methods will be investigated to confirm the efficiency of this procedure. [Copyright &y& Elsevier]

Abstract: The sinc-collocation method is presented for solving a nonlinear system of second-order boundary value problems. Some properties of the sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of the system of second-order boundary value problems to some algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results. [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]

Abstract: Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter in the quasilinear parabolic equation , in with known initial and boundary conditions and subject to an additional condition in the form of which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method. [Copyright &y& Elsevier]

In this work, the Adomian decomposition method is used to find the solution to a one-dimensional quasi-linear parabolic partial differential equation with a time-dependent unknown function. A wide class of physical phenomena is modelled by non-classical parabolic initial-boundary value problems. Thus the theoretical behaviour and numerical approximation of these problems have been active areas of research. The decomposition procedure first proposed by the American mathematician G. Adomian (1923-1996) is useful for obtaining both exact solutions to, and numerical approximations of, various kinds of linear and nonlinear problem. The Adomian decomposition method, which accurately computes the series solution, is of great interest in science and engineering. It provides a solution as a convergent series with components that can be elegantly computed. Sufficient conditions for convergence and stability of the approximate solution are given and the results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the global stability of difference equation,where the initial conditions y −k ,…, y −1, y 0 are non-negative, k ∈{1,2,3,…}, and the parameters p, q and r are non-negative. We show that when r < q +1, then the unique positive equilibrium of this equation is globally asymptotically stable and when r > q +1, and k is odd, then this equation possesses unbounded solutions. [Copyright &y& Elsevier]

A numerical technique for solving a second-order nonlinear Neumann problem is presented. The authors' approach is based on semi-orthogonal B-spline wavelets. Two test problems are presented and numerical results are tabulated to show the efficiency of the proposed technique for the studied problem. [ABSTRACT FROM AUTHOR]

Abstract: Parabolic partial differential equations with non-classical boundary specifications have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of a parabolic equation with non-local conditions (on four boundaries) replacing the classical boundary specifications. Finite difference methods are given for solving this parabolic equation in two-dimensional space with non-standard boundary conditions. Several finite difference procedures are given and compared in terms of accuracy and computing time. Two fully explicit schemes, two fully implicit methods, an alternating direction implicit (ADI) formula and two explicit techniques of Saulyev are studied. The unconditional stability of the explicit procedures of Saulyev is significant. The unique advantage of the unconditionally stable implicit ADI method is that it needs only the solution of tridiagonal systems. A numerical integration procedure is employed to overcome the non-standard boundary specifications. Numerical results are given to demonstrate the efficiency and accuracy of the proposed finite difference methods. [Copyright &y& Elsevier]

Many physical phenomena are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary conditions have received much attention in the last twenty years. Most of the papers were directed to the second-order parabolic equation, particularly to the heat conduction equation. One could generically classify these problems into two types; boundary value problems with nonlocal initial conditions, and boundary value problems with nonlocal boundary conditions. We will deal here with the second type of nonlocal boundary value problems that is the solution of nonlocal boundary value problems with standard initial condition. The main difficulty in the implicit treatment of the nonlocal boundary value problems is the nonclassical form of the resulting matrix of the system of linear algebraic equations. In this paper, various approaches for the numerical solution of the one-dimensional heat equation subject to the specification of mass which have been considered in the literature, are reported. Several methods have been proposed for the numerical solution of this boundary value problem. Some remarks comparing our work with earlier work will be given throughout the paper. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in engineering models are introduced. [Copyright &y& Elsevier]

In this work we are concerned with the solution of a nonlocal boundary value problem. An approach is presented for solving the two-dimensional parabolic partial differential equation subject to integral boundary specifications. The main objective is to propose an alternative method of solution, one not based on finite difference methods or finite element schemes or spectral techniques. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume. The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example. [Copyright &y& Elsevier]

In this article, we propose a new approach for solving an initial-boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study. [ABSTRACT FROM AUTHOR]

Over the last 20 years, the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of stochastic and deterministic problems involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. Furthermore in the past, only classical boundary value problems have been considered. The parabolic partial differential equations with non-classical conditions model various physical problems. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the second-order linear parabolic partial differential equation with nonlocal boundary specifications replacing the standard boundary conditions. This scheme is employed for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval. The Adomian decomposition method provides a reliable technique that requires less work when compared with the traditional techniques. This method is used by many researchers to investigate several scientific applications. Some experimental results using the newly proposed procedure are given to confirm our belief of the reliability of the approach. [ABSTRACT FROM AUTHOR]

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\hskip1pty, t) = \vint u(x, y, t)\,\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Several second-order finite difference schemes are discussed for solving a non-local boundary value problem for two-dimensional diffusion equation with Neumann's boundary conditions. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Using a suitable transformation the solution of this problem is equivalent to the solution of two other problems. The former which is a one-dimensional non-local boundary value problem gives the value of μthrough using the unconditionally stable standard implicit (3,1) backward time centred space (denoted BTCS) scheme. Using this result the second problem will be changed to a classical two-dimensional diffusion equation with Neumann's boundary conditions which will be solved numerically by using two unconditionally stable fully implicit finite difference schemes, or using two conditionally stable fully explicit finite difference techniques. For each method investigated the modified equivalent partial differential equation is employed which permits the order of accuracy of the numerical techniques to be determined. The results of a numerical example for all finite difference schemes discussed in this paper are given and computation times are presented. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

• A new meshless method is employed to solve fractional modified distributed-order anomalous sub-diffusion equation. • The mentioned method is based on the shape functions of reproducing kernel particle method and Galekrin idea. • The stability and convergence of the developed method have been studied. • The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. • Some examples are studied that they show the efficiency of the proposed method. • The numerical results confirm the theoretical concepts. In the Galerkin weak form technique based on various kernels that they do not have δ -Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ -Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true. [ABSTRACT FROM AUTHOR]