Volterra Integral Equations Of Second Kind Research Articles

Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters.

Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK α,γ-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

This work studies the fourth-kind integral equation as a mixed system of first and second-kind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.

A class of reducible quadrature rules for the second-kind Volterra integral equations using barycentric rational interpolation

Reducible quadrature rules constitute a well-established class of direct quadrature methods for approximating solutions to Volterra integral equations. Unlike interpolatory quadrature formulae, the reducible quadrature rule can be constructed without the need for additional calculations of moments. This paper investigates the reducible quadrature rule by employing barycentric rational interpolation to solve the underlying initial value problem and analyzes its convergence and stability. It is found that these quadrature rules exhibit high-order convergence rates accompanied by extensive stability regions. Several numerical illustrations are provided to verify the theoretical results.

Study of numerical treatment of functional first-kind Volterra integral equations

<abstract><p>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.</p></abstract>

Multistep collocation technique implementation for a pantograph-type second-kind Volterra integral equation

<p>In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and analytical solutions have been shown. To explore the solvability of the system derived from the numerical method, we have defined particular operators and demonstrated that these operators are both compact and bounded. Solvability is studied by means of the innovative compact operator concepts. The concept of convergence has been examined in greater detail, revealing that the convergence of the method is influenced by the spectral radius of the matrix generated according to the collocation parameters in the difference equation resulting from the method's error. Finally, two numerical examples are given to certify our theoretically gained results. Also, since the proposed numerical method is local in nature, it can be compared to other local methods, such as those used in reference <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We will compare our method with <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> in the last section.</p>

Efficient Nyström-type method for the solution of highly oscillatory Volterra integral equations of the second kind

Highly oscillatory Volterra integral equations are frequently encountered in engineering applications. The Nyström-type method is an important numerical approach for solving such problems. However, there remains scope to further optimize and accelerate the Nyström method. This paper presents a novel Nyström-type method to efficiently approximate solutions to second-kind Volterra integral equations with highly oscillatory kernels. First, the unknown function is interpolated at Chebyshev points. Then the integral equation is solved using the Nyström-type method, which leads to a problem of solving a system of linear equations. A key contribution is the technique to express the fundamental Lagrange polynomial in matrix form. The elements of the matrix, which involves highly oscillatory integrals, are calculated by using the classical Fejér quadrature formula with a dilation technique. The proposed method is more efficient than the one proposed in the recent literature. Numerical examples verify the efficiency and accuracy of the proposed method.

The barycentric rational predictor-corrector schemes for Volterra integral equations

This paper introduces a family of barycentric rational predictor-corrector schemes based on the Floater–Hormann family of linear barycentric rational interpolants (LBRIs) for the numerical solution of classical systems of second-kind Volterra integral equations. Also, we introduce a family of LBRI-based predictor-corrector starting procedures that is essentially explicit and whose order of convergence can be as high as that of the main method. Numerical tests verify the theoretical results on the convergence order and stability and illustrate the efficiency and power of the developed family of methods in solving stiff equations.

Definition of the solution of the homogeneous Volterra integral equation

In this paper, we study the solvability of a second-kind Volterra integral equation. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not «compressiblek. Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. The solution of the homogeneous integral equation corresponding to the original nonhomogeneous integral equation found in explicit form. Special cases of a homogeneous integral equation and its solutions are written for different values of the parameter k. Classes are indicated in which the integral equation has a solution. Singular integral equations were considered in works [1–3]. Their kernels were also «incompressiblek, but kernels had an another form. In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work.

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine.

In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra-Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.

On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

Discontinuous Galerkin approximations to second-kind Volterra integral equations with weakly singular kernel

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results.

A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays

In this paper, we develop a hybrid spectral method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel and vanishing delays. Our main strategy is to divide the original interval into subintervals, to employ the shifted generalized log orthogonal functions (GLOFs) as the basis on the first interval, to take the classical shifted Legendre polynomials as the basis on other intervals. We analyze the existence and uniqueness of the numerical scheme, and derive the corresponding error estimates. A series of examples demonstrate the effectiveness of the proposed method.

New spectral element method for Volterra integral equations with weakly singular kernel

In this paper, we put forward a new spectral element method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel, which employs shifted Müntz–Jacobi functions and shifted Legendre polynomials as basis functions. This method is capable of approximating the limited regular solution more efficiently. We analyze the existence and uniqueness of the solution to the numerical scheme and derive the hp-version optimal convergence under some reasonable assumptions. A series of numerical examples are presented to demonstrate the efficiency of the new method.

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time $$t = 0$$ , only $$1 - \alpha $$ global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for $$m = 1$$ at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.

Explicit unconditionally stable methods for the heat equation via potential theory

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step $\Delta t$ is of the order $O(\Delta x^2)$, where $\Delta x$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_dT^{d/2}$ times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $\Delta t < C/\kappa^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in $L^\infty$-norm.

Collocation methods for integro-differential algebraic equations with index 1

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations

Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocatio...

Superconvergence results for linear second-kind Volterra integral equations

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.