Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

Mirzaee, Farshid; Hadadiyan, Elham

doi:10.1002/mma.4237

Cited by 21 publications

(10 citation statements)

References 13 publications

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“…Also, we obtain the error in mid‐points of each interval [ ih , ( i + 1) h ) with

e ((), x_{i}) = ∣ u ((), x_{i}) - u^{*} ((), x_{i}) ∣, i = 0, 1, \dots, m - 1,

for

x_{i} = ((), 2 i + 1) \frac{h}{2}, i = 0, 1, \dots, m - 1,

where u ( x ) and u * ( x ) are exact and approximate solutions of Volterra‐Fredholm integral equation, respectively.Example Consider the following linear Volterra‐Fredholm integral equation 13

u ((), x) = \frac{2}{3} x - \frac{1}{3} x^{4} + \int_{0}^{x} italicxtu ((), t) italicdt + \int_{0}^{1} italicxtu ((), t) italicdt,

where the exact solution is u ( x ) = x . We compare our method with the numerical results of Scaling Functions Interpolation (SFI) method 9 for this example. Example Consider the following linear Volterra‐Fredholm integral equation 5

u ((), x) = - \frac{2}{5} x^{7} - \frac{5}{4} x^{4} + x^{3} - \frac{59}{20} x + 1 + \int_{0}^{x} ((), 2 x^{2} t + 1) u ((), t) italicdt + \int_{0}^{1} x ((), t + 1) u ((), t) italicdt,

in which the exact solution is u ( x ) = x 3 + 1. For this example, we use the Repeated Trapezoidal (RT) method 5 for comparing numerical results.…”

Section: Numerical Examplesmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution ofVolterra‐Fredholmintegral equation via hyperbolic basis functions

Esmaeili

Rostami

Hooshyarbakhsh

2020

Int J Numerical Modelling

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In this paper, a new method using hyperbolic basis functions is presented to solve second kind linear Volterra‐Fredholm integral equation. In other words, our method approximates the solution of a Volterra‐Fredholm integral equation by the hyperbolic basis functions, which produce block‐pulse functions. Hence, the new method reduces the linear Volterra‐Fredholm integral equation to a system of algebraic equations. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the new method.

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“…Also, we obtain the error in mid‐points of each interval [ ih , ( i + 1) h ) with

e ((), x_{i}) = ∣ u ((), x_{i}) - u^{*} ((), x_{i}) ∣, i = 0, 1, \dots, m - 1,

for

x_{i} = ((), 2 i + 1) \frac{h}{2}, i = 0, 1, \dots, m - 1,

where u ( x ) and u * ( x ) are exact and approximate solutions of Volterra‐Fredholm integral equation, respectively.Example Consider the following linear Volterra‐Fredholm integral equation 13

u ((), x) = \frac{2}{3} x - \frac{1}{3} x^{4} + \int_{0}^{x} italicxtu ((), t) italicdt + \int_{0}^{1} italicxtu ((), t) italicdt,

u ((), x) = - \frac{2}{5} x^{7} - \frac{5}{4} x^{4} + x^{3} - \frac{59}{20} x + 1 + \int_{0}^{x} ((), 2 x^{2} t + 1) u ((), t) italicdt + \int_{0}^{1} x ((), t + 1) u ((), t) italicdt,

in which the exact solution is u ( x ) = x 3 + 1. For this example, we use the Repeated Trapezoidal (RT) method 5 for comparing numerical results.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Many different techniques have been introduced to solve (1), such as truncated Chebyshev series, 4 moving least square method and Chebyshev polynomials 5 and inverse and direct discrete fuzzy transforms and collocation technique 6 . Some of other methods are discussed in References 7‐11.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution ofVolterra‐Fredholmintegral equation via hyperbolic basis functions

Esmaeili

Rostami

Hooshyarbakhsh

2020

Int J Numerical Modelling

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“…Mirzaee and Samadyar [13] used Bernstein collocation method for solution of 2D-mixed Volterra-Fredholm IEs. Mirzaee and Hadadiyan utilized operational matrix [14] for solution of nonlinear class of mixed Volterra-Fredholm IEs, Bell polynomials [15] for solution of nonlinear Fredholm-Volterra IEs and modification of hat functions [16] for solution of Volterra-Fredholm IEs. Mirzaee and Hadadiyan [17] also found the solution of two-dimensional Volterra-Fredholm IEs.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet

Amin

Shah

Asif

et al. 2020

Heliyon

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In this article, a computational Haar wavelet collocation technique is developed for the solution of linear delay integral equations. These equations include delay Fredholm, Volterra and Volterra-Fredholm integral equations. First we transform the derived estimates for these equations. After that, we transform these estimates to a system of algebraic equations. Finally, we solve the obtained algebraic system by Gauss elimination technique. Numerical examples are taken from literature for checking the validity and convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution. The convergence rate using distinct numbers of collocation points is also calculated, which is approximately equal to 2. All algorithms for the developed method are implemented in MATLAB (R2009b) software.

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“…Therefore, the development of effective and easy‐to‐use numerical schemes for solving such equations acquires an increasing interest. While several numerical techniques have been proposed to solve many different problems (see, for instance [22–49], and references therein), there have been few research studies that developed numerical methods to solve DOFDEs (see [50–58]). The development, however, for efficient numerical methods to solve DOFDEs is still an important issue [51].…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

Maleknejad

Rashidinia

Eftekhari

2020

Numerical Methods Partial

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In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz-Legendre wavelet approximation. We derive a new operational vector for the Riemann-Liouville fractional integral of the Müntz-Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well-known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.

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Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

Cited by 21 publications

References 13 publications

Numerical solution ofVolterra‐Fredholmintegral equation via hyperbolic basis functions

Numerical solution ofVolterra‐Fredholmintegral equation via hyperbolic basis functions

Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

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