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In this study, a novel reproducing kernel Hilbert space (RKHS) method is introduced to show that high-order linear Fredholm integro-differential equations (IDEs) with variable coefficients can be transformed into ordinary differential equation (ODEs). The RKHS method constructs multiple types of RKHSs related to the given terms based on the H - H K formulation, which are utilized to determine solutions of the Fredholm IDEs. Then analytical and numerical solutions of the Fredholm IDEs with variable coefficients are obtained by an algorithm. Finally, the effectiveness and feasibility of RKHS method have been provided to confirm our theoretical findings by some numerical results and comparisons. • A novel reproducing kernel Hilbert space method is proposed to solve the high-order linear Fredholm integro-differential equations with variable coefficients, which can determine the solution space as needed. • The innovation of this method is that a high-order linear Fredholm integro-differential equation can be transformed into an ordinary differential equation with variable coefficients based on the H − H K formulation. • The flexibility of this method lies in constructing the needed reproducing kernel, which is related to the known terms of the specific Fredholm integro-differential equation, and analytical and numerical solutions can be obtained by this method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme. • In this paper, a new approach to solve numerically neutral Volterra integro-differential equation (NVIDE) has been considered. • The last works related to numerical analysis of NVIDE were concerned with collocation method, Legendre spectral method and etc. But the most important feature of our study is a finite difference scheme has been constructed on a uniform mesh. • Using composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points, we construct the finite difference scheme which have a second-order accuracy. • Therefore we obtain numerical methods which are as high accuracy as possible. [ABSTRACT FROM AUTHOR]

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed. • Extending the Lyapunov first method to systems of Volterra integral equations and Volterra integro-differential equations. • Using the logarithmic norm to obtain sufficient stability conditions for integral and integro-differential Volterra equations. • Common approach to constructing the sufficient stability conditions for integral and integro-differential Volterra equations. • Explicit sufficient conditions via kernels and coefficients of the equations. [ABSTRACT FROM AUTHOR]

In this paper, an innovative numerical methodology is introduced for solving multiple types of linear one-dimensional Fredholm integro-differential equations using least squares support vector machine (LS-SVM) algorithm, even in singular cases. The proposed method combines the original LS-SVM model with the composite Simpson method, which transforms the original solution problem into an optimization problem with equality constraints. Then the original problem is reduced to seeking the LS-SVM model parameters, which further reduces the calculation cost. An explicit expression for the numerical solution of the Fredholm integro-differential equations is given. Finally, the constructed LS-SVM model is evaluated through numerical experiments, and the obtained solutions demonstrate the robustness and validity of the constructed model. • Introduction to traditional numerical methods for solving integro-differential equations. • The concept of least squares support vector machines and its application are discussed. • Combining LS-SVM method with Simpson's orthogonal formula to transform the problem into linear equations. • Numerical experiments verify the effectiveness and superiority of our proposed method. [ABSTRACT FROM AUTHOR]

In this study, we introduce and evaluate the effectiveness of the central finite difference coupled with a composite trapezoidal rule on adapted meshes in solving a second-order singularly perturbed Fredholm integro-differential equation with a discontinuous source term. A grid equidistribution technique was employed to address the challenge posed by the perturbation parameter in classical numerical methods. Our approach demonstrated superior precision compared to existing schemes in the literature and achieved second-order uniform convergence. • New numerical method for singularly perturbed Fredholm integro-differential equation with discontinuous source. • The method uses finite difference scheme and trapezoidal rule on adapted meshes. • The adaptive mesh generated using grid equidistribution achieves superior precision. • Better numerical outcomes compared to existing methods. • Second-order uniform convergence. [ABSTRACT FROM AUTHOR]

We propose a microscopic model describing the process of DNA thermal denaturation. The process consists of the splitting of DNA base pairs, or nucleotides, resulting in the separation of two complementary DNA strands. In contrast to the previous modelling attempts we take into account the states of all base pairs of DNA which in fact imposes the microscopic nature of the approach. The model is a linear integro-differential non-autonomous equation describing the dynamics of probability density which characterizes the distances between the bases within individual base pairs. The non autonomous structure of the equation comes from the dependence on temperature that may be in the general case a (given) function of time. To our knowledge it is the first model taking into account not only the strength of double and triple hydrogen bonds between the complementary bases but also the stacking interactions between neighborhood base pairs. In the present paper the basic preliminary properties of the model including the existence, uniqueness and stability results in various cases are discussed. Moreover the symmetrized version of the model is considered and the "macroscopic limit" is studied. The performed numerical simulations reproduce the sigmoid shape of DNA melting curves and reveal the appearance of experimentally observed denaturation bubbles. [ABSTRACT FROM AUTHOR]

In this paper, we theoretically and numerically deal with nonlinear Volterra integro-differential equations with Itô integral under a one-sided Lipschitz condition and polynomially growth conditions. It is proved that both the exact solutions and vector fields are bounded and satisfy a Hölder condition in the p th moment sense. Analogously, the boundedness and Hölder condition in the p th moment sense are preserved by the semi-implicit Euler method for sufficiently small step-size. Moreover, by the local truncated errors, we prove the strong convergence order 1. Finally, numerical simulations on stochastic control models and stochastic Ginzburg–Landau equation illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm L t 1 (0 , ∞ ; L x 2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds. [ABSTRACT FROM AUTHOR]

Abstract The reproducing kernel method is applied to Volterra nonlinear integro-differential equations. In this technique, the nonlinear term is replaced by its Taylor series. The exact solution is represented in the form of series in the reproducing Hilbert kernel space. The approximation solution is expressed by n -term summation of reproducing kernel functions. Some numerical examples are solved in two different spaces and parameters of n. Measurements of the experimental data is an indications of stability and convergence on the reproducing kernel. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a fast and efficient numerical method to solve the two-dimensional integro-differential equation with a weakly singular kernel. The numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit (ADI) method in time, combined with the second-order fractional quadrature convolution rule introduced by Lubich and the classical L 1 approximation for Caputo fractional derivative. The detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order O (τ min { 1 + α , 2 − α } + h 1 2 + h 2 2). Some numerical results are also given to confirm our theoretical prediction. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix. [ABSTRACT FROM AUTHOR]

Abstract The controllability of an impulsive process governed by a parametric integro-differential equation involving a Volterra operator is shown. The model is studied via the existence of impulsive mild solutions for a semilinear integro-differential inclusion. A discussion on the impulse functions is presented. [ABSTRACT FROM AUTHOR]

Highlights • An improved Lagrange polynomial are provided for solving multi dimensional systems of Fredholm integral equations. • The technique can improve the common disadvantages of the Lagrange interpolation, which is stable with special nodal distribution. • Theoretical analysis and convincible numerical examples are provided. • The method can easily extend to solve multi dimensional integro-differential problems. Abstract A highly accurate meshfree approach based on the barycentric Lagrange basis functions is reported for solving the linear and nonlinear multi-dimensional systems of Fredholm integral equations (FIEs) of the second kind. The method is an improved Lagrange interpolation technique which possesses high precision and inexpensive procedure. The systems of FIEs are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations. The stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric Lagrange formula, which is dominated by barycentric weights. The convergence analysis and error estimation are included. Some computational results indicate that the method is simple and efficient. [ABSTRACT FROM AUTHOR]

Abstract We study a kinetic equation which describes self-organization of various complex systems, assuming the interacting rate with small support. This corresponds to interactions between an agent with a given internal state and agents having short distance states only. We identify all possible stationary (equilibrium) solutions and describe the possibility of creating of bipolar (bimodal) distribution that is able to capture interesting behavior in modeling systems, e.g. in political sciences. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time-fractional Swift–Hohenberg equation in the sense of Riemann–Liouville derivative. Via Fourier method, the method is convergent and unconditionally stable. Also, we investigated the existence and uniqueness of the proposed method. Numerical results are demonstrated to validate the applicability and the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science, which means increased attention to fractional calculus. This paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in Hilbert space within appropriate kernel functions. The algorithm relies on the reproducing kernel Hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the Fourier coefficients of orthogonalization process. The Caputo fractional derivatives are introduced to address these issues. Moreover, the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions. The superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples. The outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method. [ABSTRACT FROM AUTHOR]

• A balanced space-time Sinc-collocation method for viscoelastic heat. • An effective single exponential Sinc method to deal with the initial singularity. • The error estimate is proved rigorously. • Space-time spectral-order convergence order simultaneously. The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems. [ABSTRACT FROM AUTHOR]

Abstract General nonlinear Volterra integro-differential equations are considered. Explicit criteria for uniform asymptotic stability and exponential stability of such equations are given. Applications to models of growth of biological populations and of grazing systems are presented. [ABSTRACT FROM AUTHOR]

Abstract Recently, Njeru and Guo presented an accelerated SOR-like (ASOR) method for solving the large and sparse augmented systems. In this paper, we establish an accelerated symmetric SOR-like (ASSOR) method, which is an extension of the ASOR method. Furthermore, the convergence properties of the ASSOR method for augmented systems are studied under suitable restrictions, and the functional equation between the iteration parameters and the eigenvalues of the relevant iteration matrix is established in detail. Finally, numerical examples show that the ASSOR is an efficient iteration method. [ABSTRACT FROM AUTHOR]

Abstract We investigate the existence of solutions for Caputo type sequential fractional integro-differential equations and inclusions subject to nonlocal boundary conditions involving Riemann–Liouville and Riemann–Stieltjes integrals. For the proofs of our main theorems we use the contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators in the case of fractional equations, and the nonlinear alternative of Leray–Schauder type for Kakutani maps and the Covitz–Nadler fixed point theorem in the case of fractional inclusions. Some examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a semi-explicit form of Volterra integro-differential-algebraic equations (IDAEs) and investigate the existence and uniqueness of solution of these systems by using differentiability index. Numerical method based on discontinuous piecewise polynomial approximation is proposed for the solution of the semi-explicit IDAEs and global convergence results are established. The performance of the numerical scheme is illustrated by means of some test problems. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples. [ABSTRACT FROM AUTHOR]

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of mild solutions of Hilfer fractional stochastic integro-differential equations with nonlocal conditions. The main results are obtained by using fractional calculus, semigroups and Sadovskii fixed point theorem. In the end, an example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

System of Maxwell equations is considered. Reduction to the integro-differential form is given. Existence, uniqueness and large time behavior of solutions of the initial-boundary value problem for integro-differential model with two-component and one-dimensional case are studied. Finite difference scheme is investigated. Wider class of nonlinearity is studied than one has been investigated before. FreeFem++ realization code and results of numerical experiments are given. [ABSTRACT FROM AUTHOR]

A Sinc-Nyström method for Volterra integro-differential equations was developed by Zarebnia (2010). The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: (1) the regularity of the solution is analyzed, and (2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided. [ABSTRACT FROM AUTHOR]

In this paper, a new method for solving coupled systems of nonlinear fractional order integro-differential equations is proposed. The idea is to use Bernoulli wavelets and operational matrix. The main purpose of the technique is to transform the studied systems of fractional order integro-differential equations into systems of algebraic equations which can be solved easily. Illustrative examples and comparisons with Haar wavelets and Legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis. [ABSTRACT FROM AUTHOR]

• We propose a new model unifying local and stochastic volatilities and Levy type of jumps. • We use asymptotics and Fourier analysis to obtain a formula for European option prices. • Our model outperforms two benchmark models for short time-to-maturity European options. • Our model outperforms the Carr-Wu model in view of the time cost of pricing European options. We introduce a new mixed model unifying local volatility, pure stochastic volatility, and L e ´ vy type of jumps in this paper. Our model framework allows the pure stochastic volatility and the jump intensity to be functions of fast (and slowly) varying stochastic processes and the local volatility to be given in a constant elasticity of variance type of parametric form. We use asymptotic analysis to derive a system of partial integro-differential equations for the prices of European derivatives and use Fourier analysis to obtain an explicit formula for the prices. Our result is an extension of a stochastic-local volatility model from no jumps to L e ´ vy jumps and an extension of a multiscale stochastic volatility model with L e ´ vy jumps from the zero elasticity of variance to the non-zero elasticity of variance. We find that our model outperforms two benchmark models in view of fitting performance when the time-to-maturities of European options are relatively short and outperforms one benchmark model in terms of pricing time cost. [ABSTRACT FROM AUTHOR]

This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time a p r i o r i error bounds. To handle the integral term, an extended Ritz-Volterra projection is introduced, which helps in achieving optimal order convergence of O (h k + 1) for the semi-discrete problem when polynomials of degree k ≥ 0 are used to approximate both the solution and the flux variables. Further, element-by-element post-processing is proposed, and it is established that it achieves convergence of the order O (h k + 2) for k ≥ 1. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results. [ABSTRACT FROM AUTHOR]

This paper contributes an efficient numerical approach for solving the systems of high-order linear Volterra integro-differential equations with variable coefficients under the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Fibonacci polynomials and their derivatives by means of collocations. In addition, the method is presented with error. Numerical results with comparisons are given to demonstrate the applicability, efficiency and accuracy of the proposed method. The results of the examples indicated that the method is simple and effective, and could provide an approximate solution with high accuracy or exact solution of the system of high-order linear Volterra integro-differential equations. [ABSTRACT FROM AUTHOR]

We prove that a second order backward difference semi-discretization approximation for the integro-differential equations with the multi-term kernels preserves the weighted asymptotic integrabilities of continuous solutions. The method of proof extend and simulate numerically those introduced by Hannsgen and Wheeler, relying on deep frequency domain techniques. [ABSTRACT FROM AUTHOR]

A Simple algorithm is used to achieve exact solutions of systems of linear and nonlinear integro- differential equations arising in many scientific and engineering applications. The algorithm does not need to find the Adomain Polynomials to overcome the nonlinear terms in Adomain Decomposition Method (ADM). It does not need to create a homotopy with an embedding parameter as in Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM). Unlike VIM, it does not need to find Lagrange Multiplier. In this manuscript no restrictive assumptions are taken for nonlinear terms. The applied algorithm consists of a single series in which the unknown constants are determined by the simple means described in the manuscript. The outcomes gained by this algorithm are in excellent concurrence with the exact solution and hence proved that this algorithm is effective and easy. Four systems of linear and nonlinear integro-differential equations are solved to prove the above claims and the outcomes are compared with the exact solutions as well as with the outcomes gained by already existing methods. [ABSTRACT FROM AUTHOR]

The integro-differential population balance equation describing aggregation processes was proposed almost 100 years ago. Aggregation is an important size enlargement process in many industries; the modeling and design of the process can be done using the population balance framework, however it is typically impossible to obtain analytical solutions: in almost every case a numerical solution of the equations must be obtained. In this paper, we present the developed group analysis method for the one-dimensional population balance equation for aggregation in a well-mixed batch system including a crystal growth term. The determining equations are solved, the optimal system, invariant solutions and all the reduced equations are obtained. Furthermore, finding the determining equation by use of the preliminary group classification is also considered. [ABSTRACT FROM AUTHOR]

In this article, we present a numerical method to numerically solve a time-fractional convection–diffusion equation. Our method is based on the operational matrices of shifted Jacobi polynomials. At first, problem is converted to a homogeneous problem by interpolation and afterward an integro-differential equation is yielded. Then we approximate the known and unknown functions with the help of shifted Jacobi functions. A system of nonlinear algebraic equations is obtained. Finally, the unknown coefficients are determined by Mathematica TM . We implemented the proposed method for several examples that they indicate the high accuracy method. It should be noted that this method is generalizable to some appropriate problems. [ABSTRACT FROM AUTHOR]

A splitting implicit-explicit (SIMEX) scheme for solving a partial integro-differential Fokker–Planck equation related to a jump-diffusion process is investigated. This scheme combines the Chang–Cooper method for spatial discretization with the Strang–Marchuk splitting and first- and second-order time discretization methods. It is proved that the SIMEX scheme is second-order accurate, positive preserving, and conservative. Results of numerical experiments that validate the theoretical results are presented. [ABSTRACT FROM AUTHOR]

Volterra equations governed by oscillatory functions arise frequently in applied fields. To construct efficient algorithms for solving these equations, oscillatory properties of their solutions should be studied. In this paper, oscillatory orders of solutions to a class of highly oscillatory Volterra integral equations are presented. Then some modified solutions are given with the help of asymptotic expansions of highly oscillatory integrals. Finally, theoretical analysis verifies these modified solutions enjoy less oscillation than the original one, and extensions to highly oscillatory Volterra integro-differential equations are considered. [ABSTRACT FROM AUTHOR]

• Bivariate Mittag-Leffler functions are classified into broad families. • All the studied functions emerge as solutions of fractional integro-differential equations. • Special cases whose importance has already been seen in the literature. • Indications for future extensions to trivariate and multivariate Mittag-Leffler functions. The various bivariate Mittag-Leffler functions existing in the literature are gathered here into two broad families. Several different functions have been proposed in recent years as bivariate versions of the classical Mittag-Leffler function; we seek to unify this field of research by putting a clear structure on it. We use our general bivariate Mittag-Leffler functions to define fractional integral operators (which have a semigroup property) and corresponding fractional derivative operators (which act as left inverses and analytic continuations). We also demonstrate how these functions and operators arise naturally from some fractional partial integro-differential equations of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of a new class of multi-valued fractional impulsive stochastic partial integro-differential equations with infinite delay in Hilbert spaces is studied. Firstly, by using stochastic analysis, analytic α -resolvent operator, fractional powers of closed operators and a fixed point theorem for multi-valued maps, we prove an existence result of mild solutions for the control systems under the mixed Lipschitz and Carathéodory conditions. Secondly, we discuss a new set of sufficient conditions for the approximate controllability of the systems. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the solution for two-dimensional hyperbolic integro-differential equations subject to purely integral conditions. First, it is demonstrated the existence and uniqueness of the solution under certain conditions. For the numerical approach, a Galerkin method based on least squares is proposed. The numerical examples illustrate the technique and show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the convergence of numerical methods to solve initial value problems of neutral delay integro-differential equations (NDIDEs). The error estimation of Runge–Kutta methods for a class of nonlinear systems of NDIDEs are derived. Some numerical experiments are given which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of solutions for impulsive fractional q -integro-difference equations with separated boundary conditions. Existence results are proved via fixed point theorems due to Krasnoselskii and O’Regan, while the uniqueness of solutions is accomplished by means of Banach’s contraction mapping principle. Examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, based on the second Chebyshev wavelets (SCW) operational matrix of fractional order integration, a numerical method for solving a class of fractional integro-differential equations with a weakly singular kernel is proposed. By using the operational matrix, the fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations. The upper bound of the error of the second Chebyshev wavelets expansion is investigated. Finally, some numerical examples are shown to illustrate the efficiency and accuracy of the approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of spatial dynamics of a class of partially sedentary integro-difference population models in a periodic habitat. For the general case of recruitment functions, we establish the existence and computation formula of spreading speeds. It is shown that the spreading speed is linearly determinate and coincides with the minimal wave speed of periodic traveling waves. Some effects of dispersal kernels, spatially variations and dispersal probabilities on spreading speeds are also investigated. [ABSTRACT FROM AUTHOR]

In this article, we deal with a system of linear second order Volterra integro-differential equations of the second kind and their numerical solutions using Single Term Walsh Series (STWS) technique. The given system of linear second order Volterra integro-differential equations of the second kind is converted into a linear system of algebraic equations. Solving this linear system of algebraic equations, we find the discrete solutions to the system of linear second order Volterra integro-differential equations of the second kind. Numerical examples are presented and the numerical results are compared with the one of the spectral methods, Chebyshev polynomial method to show the efficiency of the STWS method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new local construction of integro quartic splines which successfully works without any additional end conditions and without solving any system of equations. In fact, we obtain analytical formulae for values of the integro spline and up to its third-order derivatives values at the knots. Numerical experiments show our method is easy to implement and effective as well. [ABSTRACT FROM AUTHOR]

In this article, the Legendre spectral collocation method has been applied to solve Fredholm integro-differential-difference equations with variable coefficients. The proposed method is based on the Gauss–Legendre points with the basis functions of Lagrange polynomials. Usually, this type of integral equations are very difficult to solve analytically as well as numerically. The presented method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by Legendre spectral collocation method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we have a tendency to implement different fixed point theorem [ Banach contraction principle, Krasnoselskii’s [18] and Schaefer’s [, 18] coupled with solution operator to analyze the existence and uniqueness results for an impulsive fractional integro-differential equations (IFIDE) with state-dependent delay (SDD) in Banach spaces. Finally, cases are offered to demonstrate the concept. [ABSTRACT FROM AUTHOR]

In this article, we develop the Chebyshev wavelet method for solving the fractional delay differential equations and integro-differential equations. According to the development, we approximate the delay unknown functions by the Chebyshev wavelets series at delay time, which we call the delay Chebyshev wavelet series. We also proposed a technique by combining the method of steps and Chebyshev wavelet method for solving fractional delay differential equations. This technique converts the fractional delay differential equation on a given interval to a fractional non-delay differential equation over that interval, by using the function defined on previous interval, and then apply the Chebyshev wavelet method on the obtained fractional non-delay differential equation to find the solution. Numerical examples will be presented to demonstrate the benefits of computing with these approaches. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide new higher-order pseudospectral integration matrices (HPIMs) for the Chebyshev-type points, and present an exact, efficient, and stable approach for computing the HPIMs. The essential idea is to reduce the computation of HPIMs to that of higher-order Chebyshev integration matrices (HCIMs), and take a very simple and recursive way to compute the HCIMs efficiently and stably. Extensive numerical results show that the new approach for computing the HPIMs has better stability than that of the recently derived Elgindy’s approach for large number of Chebyshev-type points. [ABSTRACT FROM AUTHOR]

In this study, a numerical matrix method based on Chelyshkov polynomials is presented to solve the linear functional integro-differential equations with variable coefficients under the initial-boundary conditions. This method transforms the functional equation to a matrix equation by means of collocation points. Also, using the residual function and Mean Value Theorem, an error analysis technique is developed. Some numerical examples are performed to illustrate the accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]