Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels

2019 ◽  
Vol 347 ◽  
pp. 149-161 ◽  
Author(s):  
Yubin Pan ◽  
Jin Huang ◽  
Yanying Ma
Keyword(s):  
Integral Equations ◽  
Fractional Order ◽  
Volterra Integral Equations ◽  
Series Solutions ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear And Nonlinear ◽  
Multidimensional Linear

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.

scholarly journals Singularly perturbed Volterra integral equations with weakly singular kernels

2002 ◽  
Vol 30 (3) ◽  
pp. 129-143 ◽  
Author(s):  
Angelina Bijura
Keyword(s):  
Integral Equations ◽  
Special Function ◽  
Initial Boundary ◽  
Volterra Integral Equations ◽  
Singularly Perturbed ◽  
Interesting Aspect ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

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