scholarly journals On Weakly Singular Versions of Discrete Nonlinear Inequalities and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Kelong Cheng ◽  
Chunxiang Guo ◽  
Qingke Zeng
Keyword(s):  
Difference Equations ◽  
Upper Bound ◽  
Nonlinear Difference Equations ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.

scholarly journals Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Kelong Zheng ◽  
Wenqiang Feng ◽  
Chunxiang Guo
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Upper Bound ◽  
Singular Kernel ◽  
Nonlinear Difference Equation ◽  
Volterra Type ◽  
Weakly Singular Kernel ◽  
Weakly Singular

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

scholarly journals On multi-singular integral equations involving n weakly singular kernels

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1323-1333 ◽  
Author(s):  
Sales Nabavi ◽  
O. Baghani
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Singular Integral ◽  
Applied Mathematics ◽  
Uniform Space ◽  
Singular Integral Equations ◽  
Restrictive Assumption ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

We deal with some sources of Banach spaces which are closely related to an important issue in applied mathematics i.e. the problem of existence and uniqueness of the solution for the very applicable weakly singular integral equations. In the classical mode, the uniform space (C[a,b], ||.||?) is usually applied to the related discussion. Here, we apply some new types of Banach spaces, in order to extend the area of problems we could discuss. We consider a very general type of singular integral equations involving n weakly singular kernels, for an arbitrary natural number n, without any restrictive assumption of differentiability or even continuity on engaged functions. We show that in appropriate conditions the following multi-singular integral equation of weakly singular type has got exactly a solution in a defined Banach space x(t) = ?p,i=1 ?i/?(^?i) ?t,0 fi(s,x(s)) (tn-tn-1)1-?i,n...(t1-s)1-?i,1 dt + ?(t). In particular we consider the famous fractional Langevin equation and by the method we could extend the region of variations of parameter ?+ ? from interval [0,1) in the earlier works to interval [0,2).

scholarly journals High Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second Kind

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Lu Pan ◽  
Xiaoming He ◽  
Tao Lü
Keyword(s):  
Differential Equations ◽  
Computational Cost ◽  
Posteriori Error ◽  
High Accuracy ◽  
Combination Method ◽  
Weakly Singular Kernels ◽  
A Posteriori Error ◽  
Weakly Singular ◽  
Accuracy Order ◽  
Singular Kernels

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.

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