Application of two-dimensional hat functions for solving space-time integral equations

Mirzaee, Farshid; Hadadiyan, Elham

doi:10.1007/s12190-015-0915-5

Cited by 9 publications

(6 citation statements)

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“…Example Consider the following three‐dimensional nonlinear mixed Volterra–Fredholm integral equation

u (x, y, z) = f (x, y, z) + \frac{1}{2} \int_{0}^{x} \int_{0}^{1} \int_{0}^{1} x^{2} t (y z + s r) u^{3} (s, t, r) d r d t d s;

where ( x , y , z )∈ D and

f (x, y, z) = y z \sin x + \frac{1}{1800} x^{2} (4 \sin (x) (\cos^{2} (x) - 7) + 12 x \cos (x) (3 - \cos^{2} (x)) + 15 y z \cos (x) (3 - \cos^{2} (x)) - 30 y z) .

The exact solution is u ( x , y , z ) = y z sin( x ). The error results for this example is illustrated in Table along with the comparison of the error computed for the selected grid points by the present method and hat functions method . In Figures , we give the comparison of absolute error for present method and hat functions method for m = 4,6,8, respectively.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The exact solution is u ( x , y , z ) = x 2 y z . The error results for this example is illustrated in Table along with the comparison of the error computed for the selected grid points by the present method and hat functions method . In Figures , we give the comparison of absolute error for present method and hat functions method for m = 4,6,8, respectively.…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Mirzaee

Hadadiyan

2016

Math Methods in App Sciences

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This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three‐dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Example Consider the following three‐dimensional nonlinear mixed Volterra–Fredholm integral equation

u (x, y, z) = f (x, y, z) + \frac{1}{2} \int_{0}^{x} \int_{0}^{1} \int_{0}^{1} x^{2} t (y z + s r) u^{3} (s, t, r) d r d t d s;

where ( x , y , z )∈ D and

f (x, y, z) = y z \sin x + \frac{1}{1800} x^{2} (4 \sin (x) (\cos^{2} (x) - 7) + 12 x \cos (x) (3 - \cos^{2} (x)) + 15 y z \cos (x) (3 - \cos^{2} (x)) - 30 y z) .

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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

Mirzaee

Hadadiyan

2016

Math Methods in App Sciences

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“…The solution of stochastic Ito-Volterra integral equations based on stochastic operational matrix [11], E. Babolian et al have applied this method for solving systems of nonlinear integral equations [5], M. H. Heydari et al have applied Hat functions for solving nonlinear stochastic Ito integral equations [11,13]. F. Mirzaee and E. Hadadiyan have used two-dimensional Hat functions for solving space-time integral equations [17]. M. P. Tripathi et al have applied HFs for solving fractional differential equations [27].…”

Section: Introductionmentioning

confidence: 99%

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

Khajehnasiri¹,

Kermani²,

Ezzati³

2021

KgJMat

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“…Quadratic integral equations always arise in many problems of mathematical physics and chemical such as theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory and many other applications. Existence solution and numerical method to solve these type of integral equations have been studied in previous papers …”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of stochastic quadratic integral equations via operational matrix method

Mirzaee

Samadyar

2018

Math Methods in App Sciences

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In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h 2 ). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method. KEYWORDSBrownian motion process, error analysis, operational matrix, quadratic integral equations, stochastic integral equations | INTRODUCTIONThe various kinds of integral equations are important mathematical tools for describing knowledge models that appear in different areas of applied science. Because of extensive application of integral equations and not having the exact solutions in many cases, numerical solution of integral equations has attracted researcher's attention to develop numerical method for approximating solution of these equations. Among these methods, we refer to wavelet method, 1-4 homotopy perturbation method, 5-8 collocation method, 9,10 and meshless method. 11-13 A novel algorithm to get approximate solution of these equations is to express the solution as linear combination of orthogonal or nonorthogonal basis functions and polynomials such as block-pulse functions, 14,15 hat functions, 16,17 Bernoulli polynomials, 18 Legendre polynomials, 19 Bessel polynomials, 20 Chebyshev polynomials, 21 Fibonacci polynomials, 22 and orthonormal Bernstein polynomials. 23 Mathematical modeling of various phenomena in real world is leaded to a special kind of integral equations named quadratic integral equations. Quadratic integral equations always arise in many problems of mathematical physics and chemical such as theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory and many other applications. Existence solution and numerical method to solve these type of integral equations have been studied in previous papers. [24][25][26][27][28] Recently, with increasing computational power, it becomes feasible to utilize more accurate equations to model some problems that are often dependent on a noise source. So modeling these problems naturally require the use of

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Application of two-dimensional hat functions for solving space-time integral equations

Cited by 9 publications

References 40 publications

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

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

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

On the numerical solution of stochastic quadratic integral equations via operational matrix method

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