scholarly journals AN EXTENSION OF THE PRODUCT INTEGRATION METHOD TO L1 WITH APPLICATIONS IN ASTROPHYSICS

2016 ◽  
Vol 21 (6) ◽  
pp. 774-793 ◽  
Author(s):  
Laurence Grammont ◽  
Mario Ahues ◽  
Hanane Kaboul
Keyword(s):  
Integral Equation ◽  
Existence And Uniqueness ◽  
Integration Method ◽  
Sufficient Conditions ◽  
Iterative Refinement ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Product Integration Method

A Fredholm integral equation of the second kind in L1([a, b], C) with a weakly singular kernel is considered. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0 ([a, b], C) to apply it in L1 ([a, b], C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refinement schemes which we compare one to each other. Numerical evidence is given with an application in Astrophysics.

An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ahlem Nemer ◽  
Hanane Kaboul ◽  
Zouhir Mokhtari
Keyword(s):  
Integral Equation ◽  
Integration Method ◽  
New Product ◽  
Linear Interpolation ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Product Integration Method ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

Abstract In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion

scholarly journals Numerical Techniques for Solving Linear Volterra Fractional Integral Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Safa’ Hamdan ◽  
Naji Qatanani ◽  
Adnan Daraghmeh
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Fractional Integral ◽  
Integration Method ◽  
Haar Wavelet ◽  
Numerical Techniques ◽  
Product Integration ◽  
Fractional Integral Equation ◽  
Product Integration Method ◽  
Good Agreement

Two numerical techniques, namely, Haar Wavelet and the product integration methods, have been employed to give an approximate solution of the fractional Volterra integral equation of the second kind. To test the applicability and efficiency of the numerical method, two illustrative examples with known exact solution are presented. Numerical results show clearly that the accuracy of these methods are in a good agreement with the exact solution. A comparison between these methods shows that the product integration method provides more accurate results than its counterpart.

scholarly journals On the existence and uniqueness of the solution of a nonlinear integral equation

2019 ◽  
Vol 488 (6) ◽  
pp. 595-598
Author(s):  
M. V. Nikolaev ◽  
A. A. Nikitin
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Integral Operator ◽  
Spatial Model ◽  
Existence And Uniqueness ◽  
Nonlinear Integral Equation ◽  
Sufficient Conditions ◽  
Biological Communities ◽  
Uniqueness Of The Solution

In this paper we study the nonlinear integral equation that arose in the spatial model of biological communities developed by Austrian scientists Ulf Dieckmann and Richard Law. Sufficient conditions for the existence of the solution of this equation (the fixed point of the integral operator) were found. The question of uniqueness of the solution is also studied.

scholarly journals On the functional-integral equation of Volterra type with weakly singular kernel

2008 ◽  
Vol 83 (97) ◽  
pp. 57-63
Author(s):  
Aldona Dutkiewicz
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Sufficient Conditions ◽  
Functional Integral ◽  
Singular Kernel ◽  
Volterra Type ◽  
Weakly Singular Kernel ◽  
Measures Of Noncompactness ◽  
Weakly Singular

We give sufficient conditions for the existence of Lp-solution of a Volterra functional-integral equation in a Banach space. Our assumptions and proofs are expressed in terms of measures of noncompactness.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

scholarly journals Existence Theory and Novel Iterative Method for Dynamical System of Infectious Diseases

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Gauhar Ali ◽  
Ghazala Nazir ◽  
Kamal Shah ◽  
Yongjin Li
Keyword(s):  
Dynamical System ◽  
Existence And Uniqueness ◽  
Integral Transform ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Qualitative Theory ◽  
Existence Theory ◽  
Uniqueness Of The Solution ◽  
Respective Theory

This manuscript is devoted to investigate qualitative theory of existence and uniqueness of the solution to a dynamical system of an infectious disease known as measles. For the respective theory, we utilize fixed point theory to construct sufficient conditions for existence and uniqueness of the solution. Some results corresponding to Hyers–Ulam stability are also investigated. Furthermore, some semianalytical results are computed for the considered system by using integral transform due to the Laplace and decomposition technique of Adomian. The obtained results are presented by graphs also.

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