Lévy-driven stochastic Volterra integral equations with doubly singular kernels: existence, uniqueness, and a fast EM method

2020 ◽  
Vol 46 (2) ◽  
Author(s):  
Xinjie Dai ◽  
Aiguo Xiao
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Em Method ◽  
Singular Kernels

Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations

2006 ◽  
Vol 6 (4) ◽  
pp. 436-442 ◽  
Author(s):  
A.N. Tynda
Keyword(s):  
Numerical Methods ◽  
Integral Equations ◽  
Numerical Algorithms ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Fredholm Integral Equations ◽  
Optimal Complexity ◽  
Weakly Singular ◽  
Different Types ◽  
Singular Kernels

AbstractIn this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.

scholarly journals NUMERICAL SOLUTION OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS WHICH MAY HAVE A BOUNDARY SINGULARITY

2009 ◽  
Vol 14 (1) ◽  
pp. 79-89 ◽  
Author(s):  
Marek Kolk ◽  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Piecewise Polynomial ◽  
Boundary Singularity ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Linear Volterra Integral Equations ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. Global convergence estimates are derived and a collection of numerical results is given.

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

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2105
Author(s):  
Elham Hashemizadeh ◽  
Mohammad Ali Ebadi ◽  
Samad Noeiaghdam
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Group Symmetry ◽  
Algebraic Equations ◽  
Weakly Singular Kernels ◽  
Euler’S Method ◽  
Genocchi Polynomials ◽  
Weakly Singular ◽  
Singular Kernels

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 of the proposed method.

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