Numerical Solution of Volterra Integral Equations of Second Kind (original) (raw)

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Abstract

In the present paper, we solve numerically Volterra integral equations of second kind with regular and singular kernels by given a numerical algorithm to solve the equation. Numerical example are considered to verify the effectiveness of the proposed derivations and numerical solution is compared with the existing method available in the literature.

A New Analytical Approach to First Kind Volterra Integral Equations

Research Square (Research Square), 2023

A significant group of mathematical issues known as Volterra integral equations appear in numerous scientific and engineering applications. These equations are inherently nonlinear and complicated, making analytical solution difficult. In order to solve Volterra integral equations of the first class, a novel analytical technique known as the Hussein Jassim Method (HJ-method) is presented in this research work. The Volterra integral equations are introduced mathematically in the first section of the study, which also emphasizes their importance in simulating real-world processes. The novel method's algorithm is then described, followed by an analysis. We also research the novel method's convergence characteristics and the circumstances under which it produces trustworthy results. We provide a number of Volterra integral equation examples to show the potency of this approach. In the end, the new method has proven to be effective and efficient in solving first-kind Volterra integral equations. The method has produced satisfactory results, with the obtained solution closely approximating the exact solution. These findings indicate the capability of this method to address the mathematical and technical challenges associated with solving Volterra integral equations. This also makes it possible to use the new technique in other relevant practical contexts.

Recent advances in the numerical solution of Volterra integral equations

2020

Natural Volterra Runge--Kutta methods and general linear methods are two large family of the methods which have recently attracted more attention in the numerical solution of Volterra integral equations. The purpose of the paper is the presentation of some recent advances in these methods. Also, implementation issues for these methods will be discussed.

An explicit numerical algorithm to the solution of Volterra integral equation of the second kind

2019

This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 g(t-\tau)y(\tau)\,d\tau \end{align*} with a kernel ggg and input fff for yyy. In some applications we have a smooth integrable kernel but the input fff could be a generalised function, which could involve the Dirac distribution. We call the case when f=deltaf=\deltaf=delta, the Dirac distribution centred at 0, the fundamental solution EEE, and show that E=delta+hE=\delta+hE=delta+h where hhh is integrable and solve \begin{align*} h(t)=g(t)+\int^t_0 g(t-\tau)h(\tau)\,d\tau \end{align*} The solution of the general case is then \begin{align*} y(t)=f(t)+(h*f)(t) \end{align*} which involves the convolution of hhh and fff. We can approximate ggg to desired accuracy with piecewise constant kernel for which the solution hhh is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.

A numerical method for the weakly singular Volterra integral equations

In this paper the method that presented by L.Tao and H.Young [1] for the nonlinear weakly singular Volterra integral equations of the second kind , have been considered and The convergence and error estimation have been presented. Then this method have been developed for the system of weakly singular Volterra integral equations. A convergence theorem and an error estimation is given.

A new approach to the numerical solution of Volterra integral equations by using Bernstein's approximation

Communications in Nonlinear …, 2011

In this paper, we present a numerical method for solving Volterra integral equations of the second kind (VK2), first kind (VK1) and even singular type of these equations. The proposed method is based on approximating unknown function with Bernstein's approximation. This method using simple computation with quite acceptable approximate solution. Furthermore we get an estimation of error bound for this method. For showing efficiency of this method we use several examples.

The semi-explicit Volterra integral algebraic equations with weakly singular kernels: The numerical treatments

2012

This paper deals with some theoretical and numerical results for Volterra Integral Algebraic Equations (IAEs) of index-1 with weakly singular kernels. This type of equations typically has solutions whose derivatives are unbounded at the left endpoint of the interval of integration. For overcoming this non-smooth behavior of solutions, using the appropriate coordinate transformation the primary system is changed into a new IAEs which its solutions have better regularity.

The singular value expansion of the Volterra integral equation associated to a numerical differentiation problem

Journal of Mathematical Analysis and Applications, 2018

We consider the Volterra integral equation of the first kind for the derivative of a given function with one-side boundary conditions. We give a method to obtain the singular value expansion for the corresponding integral kernel. This singular value expansion can be used to give algorithms for the solution of the numerical differentiation problem. A numerical experiment shows the results obtained by a simple version of such algorithms.

Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method

Hindawi

We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). Te considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. Te explained technique is delineated with two numerical cases to demonstrate the beneht of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.

On the numerical solution of Volterra-Fredholm integral equations with logarithmic kernel using smoothing transformation

International Journal of Applied Mathematical Research, 2014

A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel. We transform the Volterra Fredholm integral equations to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials. This lead to a system of algebraic equations with Legendre or Chebychev coefficients. Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

Abstract and Applied Analysis, 2014

We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.

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References (6)

  1. Abdul J. Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons Inc., (1999).
  2. N. Saran, S. D. Sharma and T. N. Trivedi, Special Functions, Seventh edition, Pragati Prakashan, (2000).
  3. M. Rahman, Integral equations and their Applications, WITPress, USA (2007).
  4. M. M. Rahman, M . A. Hakim, M. Kamrul Hassan, M. K. Alam and L. Nowsher Ali, Numerical solution of Volterra Integral Equations of second kind with help of Chebyshev polynomials, Annals of Pure and Applied mathematics, Vol. 1, No. 2, (2012), 158-167.
  5. Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Second edition, Tata McGraw-Hill, (2007).
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A Survey of Regularization Methods of Solution of Volterra Integral Equations of the First Kind

One of the most recent mathematical concepts, the regularization methods for solving integral equations, has brought in yet another dimension into the existing problems and has helped to usher in a new body of knowledge for further consideration. Indeed, new inputs have placed other numerical methods for solving integral equations in the fastest lane, since a number of methods were analysed with respect to accuracy, convergence and stability properties. Most of the work was done on the assumptions that the kernel and the right-hand side are known without error and that the approximating equation can be exact, whereas, as is often the case, these may not be true. Consequently, the effect can be observed in the study of regularization methods for solving Volterra type integral equations of the first kind. Classical regularization methods tend to destroy the non-anticipatory (or causal) nature of the original Volterra problem because such methods typically rely on the computation of the Volterra adjoint operator. In this paper, we highlight the general concept of the regularization method of solution for Volterra integral equation of the first kind without destroying the Volterra structure.

ON A WAY FOR SOLVING VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND

International Journal of Research -GRANTHAALAYAH, 2022

There are many classes' methods for finding of the approximately solution of Volterra integral equations of the second kind. Recently, the numerical methods have been developed for solving the integral equations of Volterra type, which is associated with the using of computers. Volterra himself suggested quadrature formula for finding the numerical solution of integral equation with the variable bounders. By using some disadvantages of mentioned methods here proposed to use some modifications of the quadrature formula which have called as the multistep methods with the fractional stepsize. This method has comprised with the known methods and found some relation between constructed here methods with the hybrid methods. And also, the advantages of these methods are shown. Constructed some simple methods with the fractional stepsize, which have the degree p≤4 of the receiving results. Here is applied one of suggested methods to solve some model problem and receive results, which are corresponding to theoretical results.

Numerical Solution of Nonlinear Volterra Integral Equations of the First Kind with Convolution Kernel

Keywords Abstract Volterra integral equation of convolution type weakly singular integral equation Piecewise constant orthogonal functions Operational matrices Inversion of Laplace transform In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0, 1) to solve nonlinear Volterra integral equations of the first kind with convolution kernel without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.

Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method

Applied Mathematics and Computation, 2007

An expansion method is used for treatment of second kind Volterra integral equations system. This method gives an analytic solution for the system. The method reduces the system of integral equations to a linear system of ordinary differential equations. After constructing boundary conditions, this system reduces to a system of equations that can be solved easily with any of the usual methods. Finally, for showing the efficiency of the method we use some numerical examples.

Comparative study of numerical methods for a nonlinear weakly singular Volterra integral equation

HERMIS J. v7, 2006

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. We investigate the application of product integration methods and a detailed analysis of the Trapezoidal method is given. In order to improve the numerical results we consider extrapolation procedures and collocation methods based on graded meshes. Several examples are presented illustrating the performance of the methods.

HYBRID NUMERICAL METHODS FOR SOLVING VOLTERRA INTEGRAL EQUATION OF THE FIRST KIND

IAEME PUBLICATION, 2020

We propose hybrid numerical methods for solving Volterra integral equations of the first kind by using Leibnitz rule, Adomian decomposition method and variational iteration method. We convert Volterra integral equation of the first kind into Volterra integral equation of the second kind by using Leibnitz rule and then apply Adomian decomposition method and variational iteration method to solve the Volterra integral equation of the second kind. To illustrate the accuracy of the proposed methods, some numerical examples have been performed.

Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

Symmetry, 2020

In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority ...