scholarly journals Biorthogonal-Wavelet-Based Method for Numerical Solution of Volterra Integral Equations

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1098 ◽  
Author(s):  
Mutaz Mohammad
Keyword(s):  
Image Processing ◽  
Integral Equations ◽  
Computational Analysis ◽  
Volterra Integral Equations ◽  
Data Recovery ◽  
Matrix Equations ◽  
Biorthogonal Wavelet ◽  
Computational Theory ◽  
Processing Data ◽  
Extension Principles

Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is converted into matrix equations via these generators. We present examples of the generated pseudo-splines quasi-affine tight framelet systems. Some numerical results to validate the proposed method are presented to illustrate the efficiency and accuracy of the method.

scholarly journals EXTRACTION OF ELASTIC, VISCOUS AND INERTIAL COMPONENTS FROM THE TOTAL REACTION OF AN ADDITIONAL SUPPORT ATTACHED TO A RECTANGULAR PLATE

2021 ◽  
pp. 20-28
Author(s):  
Alexey Voropay ◽  
Pavel Yegorov
Keyword(s):  
Rectangular Plate ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Matrix Equations ◽  
Total Reaction ◽  
Plate Vibrations ◽  
System Of Matrix Equations ◽  
Plates And Shells ◽  
Additional Support ◽  
The Impact

The paper deals with a mechanical system consisting of a hinged rectangular plate and an additional viscoelastic support with considering its mass-inertia. The impact of the characteristics of additional support on the plate strained state is studied by an original approach of extracting elastic, viscous and inertial components from the total reaction. The plate is assumed to be medium thickness, elastic and isotropic. The Timoshenko hypothesis is used for deformation equations. The external non-stationary force initiates plate vibrations. The impact of the additional support is replaced by the action of three unknown independent non-stationary concentrated forces. The basic formulas for deriving system of three Volterra integral equations are proposed. The system is then solved by numerical and analytical method. By discretizing in time the system of Volterra integral equations is reduced to a system of matrix equations. The system of matrix equations is solved with using generalized Kramer’s algorithm for block matrices and Tikhonov’s regularization method. Note that the approach proposed is applicable for other objects with additional supports, such as beams, plates and shells having various boundary contour and boundary supporting. The results of computing elastic, viscous and inertial components of total reactions on the plate are given. The approach proposed is verified by matching the results of computations by two different methods, namely numerical and analytical for one total reaction and numerical for the total reaction obtained by adding elastic, viscous and inertial components.

Path-Dependent Backward Stochastic Volterra Integral Equations with Jumps

2016 ◽  
Author(s):  
Ludger Overbeck ◽  
Jasmin A.L. RRder
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Path Dependent

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

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.

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

2020 ◽  
Vol 43 (8) ◽  
pp. 5212-5233 ◽  
Author(s):  
Masoud Saffarzadeh ◽  
Mohammad Heydari ◽  
Ghasem Barid Loghmani
Keyword(s):  
Convergence Analysis ◽  
Integral Equations ◽  
Iterative Algorithm ◽  
Volterra Integral Equations
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