Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

2019 ◽  
Vol 25 (1) ◽  
pp. 1-11
Author(s):  
Seyed Mousa Torabi ◽  
Abolfazl Tari ◽  
Sedaghat Shahmorad
Keyword(s):  
Stability Analysis ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Two Dimensional ◽  
Transform Method ◽  
One Dimensional ◽  
Convergence And Stability ◽  
Differential Transform ◽  
Convergence And Stability Analysis

Abstract In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.

scholarly journals Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Reza Abazari ◽  
Adem Kılıçman
Keyword(s):  
Integral Equations ◽  
Numerical Study ◽  
Volterra Integral Equations ◽  
Differential Transform Method ◽  
The Other ◽  
Two Dimensional ◽  
Nonlinear Integral Equations ◽  
Transform Method ◽  
Engineering Sciences ◽  
Differential Transform

The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

Approximate methods of solving amplitude-phase problem for continuous signals

2021 ◽  
pp. 37-46
Author(s):  
Ilia V. Boikov ◽  
Yana V. Zelina
Keyword(s):  
Integral Equations ◽  
Collocation Methods ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Phase Problem ◽  
Spline Collocation ◽  
Approximate Methods ◽  
Nonlinear Operator Equations ◽  
One Dimensional

Amplitude and phase problems in physical research are considered. The construction of methods and algorithms for solving phase and amplitude problems is analyzed without involving additional information about the signal and its spectrum. Mathematical models of the amplitude and phase problems in the case of one-dimensional and two-dimensional continuous signals are proposed and approximate methods for their solution are constructed. The models are based on the use of nonlinear singular and bisingular integral equations. The amplitude and phase problems are modeled by corresponding nonlinear singular and bisingular integral equations defined on the numerical axis (in the one-dimensional case) and on the plane (in the two-dimensional case). To solve the constructed nonlinear singular and bisingular integral equations, spline-collocation methods and the method of mechanical quadratures are used. Systems of nonlinear algebraic equations that arise during the application of these methods are solved by the continuous method of solving nonlinear operator equations. A model example shows the effectiveness of the proposed method for solving the phase problem in the two-dimensional case.

scholarly journals Solving Three Dimensions Volterra Integral Equations (TDVIE) via a Neural Network

2019 ◽  
Vol 23 (10) ◽  
pp. 92
Author(s):  
Nahdh S. M. Al-Saif ◽  
Ameen Sh. Ameen ◽  
Ghaith Fadhil Abbas2
Keyword(s):  
Neural Network ◽  
Integral Equations ◽  
Three Dimensional ◽  
Volterra Integral Equations ◽  
Three Dimensions ◽  
Feed Forward Neural Network ◽  
Transform Method ◽  
Linear Transfer Function ◽  
Differential Transform ◽  
Hidden Layer

The aim of this paper  is present a new numerical method for solvingThree Dimensions Volterra Integral Equations using artificial neural network by design multilayer feed forward Neural Network. A multi- layers design in our proposed method consist of a hidden layer having seven hidden units. and one linear output unit. Linear Transfer function used as each unit and using Levenberg- Marquardtalgorithmtraining. Moreover, examples on three- dimensional Volterra integral equations carried out to illustrate the accuracy and the efficiency of the presented method. In addition, some comparisons among proposed method and Shifted Chebyshev Polynomials method and Reduced Differential Transform Method are presented.   http://dx.doi.org/10.25130/tjps.23.2018.176

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