A meshless approximate solution of mixed Volterra–Fredholm integral equations

This paper presents a meshless method using a radial basis function collocation scheme for numerical solution of mixed Volterra–Fredholm integral equations, where the region of integration is a non-rectangular domain. We will show that this method requires only a scattered data of nodes in the domain. It is shown that the proposed scheme is simple and computationally attractive. Applications of the method are also demonstrated through illustrative examples.     

A new computational method for solution of non-linear Volterra–Fredholm integro-differential equations

An approximation method is developed for the solution of high-order non-linear Volterra–Fredholm integro-differential (NVFID) equations under the mixed conditions. The approach is based on the orthogonal Chebyshev polynomials. The operational matrices of integration and product together with the derivative operational matrix are presented and are utilized to reduce the computation of Volterra–Fredholm integro-differential equations to a system of non-linear algebraic equations. Numerical examples illustrate the pertinent features of the method.     

Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.     

Wilson wavelets-based approximation method for solving nonlinear Fredholm–Hammerstein integral equations

Some of mathematical physics models deal with nonlinear integral equations such as diffraction problems, scattering in quantum mechanics, conformal mapping and etc. In fact, analytically solving such nonlinear integral equations is usually difficult, therefore, it is necessary to propose proper numerical methods. In this paper, an efficient and accurate computational method based on the Wilson wavelets and collocation method is proposed to solve a class of nonlinear Fredholm–Hammerstein integral equations. In the proposed method, Kumar and Sloan scheme is used. Convergence of the Wilson expansion is investigated and also the error analysis of the proposed method is proved. Some numerical examples are provided to demonstrate the accuracy and efficiency of the method.     

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.     

Homotopy perturbation transform method for nonlinear equations using He’s polynomials

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

He’s homotopy perturbation method for continuous population models for single and interacting species

He’s homotopy perturbation method is applied for obtaining approximate analytical solutions of continuous population models for single and interacting species. In comparison with existing techniques, this method is very straightforward, and the solution procedure is very simple. Also, it is highly effective in terms of accuracy and rapid convergence. Analytical and numerical studies are presented.     

Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations

In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation form in the Hilbert space \( W_{2}^{2} \left( \varOmega \right) \oplus W_{2}^{2} \left( \varOmega \right) \). Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.

     

An expansion–iterative method for numerically solving Volterra integral equation of the first kind

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion–iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy.     

A Nyström method for a class of Fredholm integral equations on the real semiaxis

A class of Fredholm integral equations of the second kind, with respect to the exponential weight function \(w(x)=\exp (-(x^{-\alpha }+x^\beta ))\), \(\alpha >0\), \(\beta >1\), on \((0,+\infty )\), is considered. The kernel k(x, y) and the function g(x) in such kind of equations,

$$\begin{aligned} f(x)-\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {d}y =g(x),\quad x\in (0,+\infty ), \end{aligned}$$

can grow exponentially with respect to their arguments, when they approach to \(0^+\) and/or \(+\infty \). We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and “truncated” Gaussian rules, recently published in Mastroianni and Notarangelo (Acta Math Hung, 142:167–198, 2014), and Mastroianni, Milovanovi? and Notarangelo (IMA J Numer Anal 34:1654–1685, 2014) respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included.

     

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

A new approach to solving systems of linear interval equations based on the generalized procedure of interval extension is proposed. This procedure is based on the treatment of interval zero as an interval centered around zero, and for this reason it is called the “interval extended zero” method. Since the “interval extended zero” method provides a fuzzy solution to interval equations, its interval representations are proposed. It is shown that they may be naturally treated as modified operations of interval division. These operations are used for the modified interval extensions of known numerical methods for solving systems of linear equations and finally for solving systems of linear interval equations. Using a well known example, it is shown that the solution obtained by the proposed method can be treated as an inner interval approximation of the united solution and an outer interval approximation of the tolerable solution, and lies within the range of possible AE-solutions between the extreme tolerable and united solutions. Generally, we can say that the proposed method provides the results which can be treated as approximate formal solutions sometimes referred to as algebraic solutions. Seven known examples are used to illustrate the method’s efficacy and advantages in comparison with known methods providing formal (algebraic) solutions to systems of linear interval equations. It is shown that a new method provides results which are close to the so-called maximal inner solutions (the corresponding method was developed by Kupriyanova, Zyuzin and Markov) and the algebraic solutions obtained by the subdifferential Newton method proposed by Shary. It is important that the proposed method makes it possible to avoid inverted interval solutions. The influence of the system’s size and number of zero entries on the results is analyzed by applying the proposed method to the Leontief input–output model of economics.     

The variational iteration method for solving systems of equations of Emden–Fowler type

In this paper, the variational iteration method (VIM) is used to study systems of linear and nonlinear equations of Emden–Fowler type arising in astrophysics. The VIM overcomes the singularity at the origin and the nonlinearity phenomenon. The Lagrange multipliers for all cases of the parameter α,α>0, are determined. The work is supported by examining specific systems of two or three Emden–Fowler equations where the convergence of the results is emphasized.     

Solution of nonlinear pull-in behavior in electrostatic micro-actuators by using He’s homotopy perturbation method

The work presented here is about the nonlinear pull-in behavior of different electrostatic micro-actuators. He’s homotopy perturbation method (HPM) is applied to solve different types of micro-actuators like Fixed–Fixed beam and Cantilever beam actuators. Simulated results are presented for further analysis. Also the obtained results compare well with the literature.     

A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems

Solving block-tridiagonal systems is one of the key issues in numerical simulations of many scientific and engineering problems. Non-zero elements are mainly concentrated in the blocks on the main diagonal for most block-tridiagonal matrices, and the blocks above and below the main diagonal have little non-zero elements. Therefore, we present a solving method which mixes direct and iterative methods. In our method, the submatrices on the main diagonal are solved by the direct methods in the iteration processes. Because the approximate solutions obtained by the direct methods are closer to the exact solutions, the convergence speed of solving the block-tridiagonal system of linear equations can be improved. Some direct methods have good performance in solving small-scale equations, and the sub-equations can be solved in parallel. We present an improved algorithm to solve the sub-equations by thread blocks on GPU, and the intermediate data are stored in shared memory, so as to significantly reduce the latency of memory access. Furthermore, we analyze cloud resources scheduling model and obtain ten block-tridiagonal matrices which are produced by the simulation of the cloud-computing system. The computing performance of solving these block-tridiagonal systems of linear equations can be improved using our method.     

On kalandiya’ method for the numerical solution of singular integral equations

For the numerical solution of one-dimensional singular integral equations with Cauchy type kernels, one can use an appropriate quadrature rule and an appropriate set of collocation points for the reduction of this equation to a system of linear equations. In this short paper, we use as collocation points the nodes of the quadrature rule and we rederive, in a more direct manner, Kalandiya’ method for the numerical solution of the aforementioned class of equations, which was originally based on a trigonometric interpolation formula. Furthermore, we test this method in numerical applications. Finally, a discussion on the accuracy of the same method is made.     

A wavelet Petrov–Galerkin method for solving integro-differential equations

In this paper, we solve integro-differential equation by using the Alpert multiwavelets as basis functions. We also use the orthogonality of the basis of the trial and test spaces in the Petrov–Galerkin method. The computations are reduced because of orthogonality. Thus the final system that we get from discretizing the integro-differential equation has a very small dimension and enough accuracy. We compare the results with [M. Lakestani, M. Razzaghi, and M. Dehghan, Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math. Probl. Eng. 2006 (2006), pp. 1–12, Article ID 96184] and [A. Ayad, Spline approximation for first-order Fredholm integro-differential equation, Stud. Univ. Babes-Bolyai. Math., 41(3), (1996), pp. 1–8] which used a much larger dimension system and got less accurate results. In [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal. 35(1) (1998), pp. 406–434], convergence of Petrov–Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree.     

The discrete collocation method for Fredholm–Hammerstein integral equations based on moving least squares method

The purpose of this paper is to investigate the discrete collocation method based on moving least squares (MLS) approximation for Fredholm–Hammerstein integral equations. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.