scholarly journals Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 583-597 ◽  
Author(s):  
Mohammed Al-Smadi
Keyword(s):  
Integral Equations ◽  
Reproducing Kernel ◽  
Approximate Solutions ◽  
Uniform Continuity ◽  
Fuzzy Integral ◽  
System Of Integral Equations ◽  
Orthonormal Bases ◽  
Reproducing Kernel Spaces ◽  
Reproducing Kernel Theory ◽  
Physical Phenomena

Integral equations under uncertainty are utilized to describe different formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for a class of integral equations under uncertainty. The scheme presented here is based upon the reproducing kernel theory and the fuzzy real-valued mappings. The solution methodology transforms the linear fuzzy integral equation to crisp linear system of integral equations. Several reproducing kernel spaces are defined to investigate the approximate solutions, convergence and the error estimate in terms of uniform continuity. An iterative procedure has been given based on generating the orthonormal bases that rely on Gram-Schmidt process. Effectiveness of the proposed method is demonstrated using numerical experiments. The gained results reveal that the reproducing kernel is a systematic technique in obtaining a feasible solution for many fuzzy problems.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

2020 ◽  
Vol 4 (1) ◽  
pp. 9
Author(s):  
Atanaska Georgieva ◽  
Snezhana Hristova
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

scholarly journals The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

2018 ◽  
Vol 15 ◽  
pp. 8070-8080 ◽  
Author(s):  
Hameeda Oda Al-Humedi
Keyword(s):  
Hilbert Space ◽  
Exact Solutions ◽  
Integral Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Reproducing Kernel Space ◽  
Linear Ordinary Differential Equations ◽  
Weakly Singular

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find.  The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity.  From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs).   The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization. 

scholarly journals A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 701-710
Author(s):  
Wei Jiang ◽  
Beibei Guo
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Numerical Method ◽  
Reproducing Kernel ◽  
Variable Order ◽  
Piecewise Polynomials ◽  
Reproducing Kernel Spaces ◽  
Reproducing Kernel Theory ◽  
Dimensional Variable ◽  
Theoretical Results

The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any ?-approximate solution is proved and an effective method for obtaining the ?-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Sign in / Sign up

Export Citation Format

Share Document