scholarly journals THE COMPUTATIONAL EFFICIENCY OF WALSH APPROXIMATION FOR TWO-DIMENSIONAL VOLTERRA INTEGRAL EQUATIONS

2011 ◽  
Vol 04 (02) ◽  
pp. 263-270 ◽  
Author(s):  
S. Anderyance ◽  
M. Hadizadeh
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Volterra Integral Equation ◽  
Computational Efficiency ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Wavelet Packets ◽  
Walsh Functions ◽  
Two Dimensional ◽  
Operational Matrix Of Integration

In this research, we give details of a new numerical method for the approximate solution of a general two-dimensional Volterra integral equation, using the discontinuous wavelet packets e.g. Walsh functions. The double Walsh approximation we have adopted utilizes a simple robust numerical scheme for approximate solution of the equations. The two-dimensional operational matrix of integration for each subinterval [Formula: see text] is explicitly constructed, where m is a power of 2. Finally the reliability and efficiency of the proposed scheme are demonstrated by some numerical results.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

scholarly journals Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

scholarly journals PIECEWISE CONSTANT SOLUTION OF NON LINEAR VOLTERRA INTEGRAL EQUATION

2014 ◽  
pp. 95-100
Author(s):  
S. CHARAN TEJA ◽  
MONIKA MITTAL
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Volterra Integral Equation ◽  
Computational Efficiency ◽  
Walsh Functions ◽  
Piecewise Constant ◽  
Block Pulse Functions ◽  
Constant Solution ◽  
Non Linear ◽  
Linear Volterra Integral Equations

In this paper, modification in the computational methods, for solving Non-linear Volterra integral equations, is presented. Here, two piecewise constant methods are considered for obtaining the solutions. The first method is based on Walsh Functions (WF) and the second method is via Block Pulse Functions (BPF). Comparison between the two methods is presented by calculating the errors vis-à-vis exact solution. Computational efficiency of BPF is established by profiling the computations for two examples with MATLAB 7.10 Profiler.

scholarly journals Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

2021 ◽  
Author(s):  
Shyam Lal ◽  
Satish Kumar
Keyword(s):  
Integral Equation ◽  
Characteristic Function ◽  
Volterra Integral Equation ◽  
Operational Matrix ◽  
New Technique ◽  
Algebraic Equations ◽  
Approximation Of Functions ◽  
Bernoulli Wavelet ◽  
A New Technique ◽  
Operational Matrix Of Integration

AbstractIn this paper, two new estimators $$ E_{2^{k-1},0}^{(1)}(f) $$ E 2 k - 1 , 0 ( 1 ) ( f ) and $$ E_{2^{k-1},M}^{(1)}(f) $$ E 2 k - 1 , M ( 1 ) ( f ) of characteristic function and an estimator $$ E_{2^{k-1},M}^{(2)}(f) $$ E 2 k - 1 , M ( 2 ) ( f ) of function of H$$\ddot{\text {o}}$$ o ¨ lder’s class $$H^{\alpha } [0,1)$$ H α [ 0 , 1 ) of order $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.

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