A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation

Dzhumabaev, D. S.; Nazarova, K. Zh.; Uteshova, Roza

doi:10.1134/s1995080220090103

Cited by 9 publications

(5 citation statements)

References 16 publications

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“…Many authors have researched various problems for these equations. To resolve a linear two-point boundary value problem for a Fredholm integro-differential equation (FIDE), a modification of the parameterization method is suggested in [15].…”

Section: Introductionmentioning

confidence: 99%

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

Amirali

Durmaz

Acar

et al. 2023

J. Phys.: Conf. Ser.

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In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

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Section: Introductionmentioning

confidence: 99%

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

Amirali

Durmaz

Acar

et al. 2023

J. Phys.: Conf. Ser.

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“…In present work we consider problem with impulse actions for nonlinear ODEs by parameterization method by Dulat Dzhumabaev [12] proposed for solving BVPs for ODEs and extended to different classes of differential equations [13][14][15][16][17][18][19][20]. We offer a modification of parameterization method for solving to problem with impulse actions for system of nonlinear ODEs.…”

Section: Introductionmentioning

confidence: 99%

A problem with impulse actions for nonlinear ODEs

Mukash

Assanova

Stanzhytskyi

2023

ijmph

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In the paper, we consider problem with impulse actions for system of nonlinear ordinary differential equations (ODEs) in the interval. For solving this problem we use a modification of parameterization method by Dulat Dzhumabaev. In the modification of the method we introduce the parameters as the values of the unknown function at the middle of the subintervals of the partition of the considered interval. The problem with impulse actions transfers to an equivalent problem system of nonlinear ODEs with parameters. Conditions for an existence of solution to the equivalent problem are obtained. Existence theorem for solutions this problem is established by one generalizes a theorem of Hadamard. We also constructed an algorithm for finding of solution to this problem. Finally, we found conditions for solvability to the problem with impulse actions for system of nonlinear ODEs in terms of special matrix composed by the initial data. This method can be applied to various types of nonlinear problems with impulse actions for ODEs.

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“…Furthermore, researchers employed fitted analytical approaches because of the difficulty of obtaining accurate solutions to these types of problems. Some of these methods are reproducing kernel Hilbert space method [ 7 ], Nyström method [ 38 ], Touchard polynomials method [ 2 ], Tau method [ 20 , 32 ], Collocation and Kantorovich methods [ 37 ], Galerkin method [ 12 , 41 , 43 ], Boole collocation method [ 14 ], parameterization method [ 17 ], Legendre collocation matrix method[ 44 ], variational iteration technique [ 19 ]. The increasing interest in recent years is not limited to only FIDEs, but also the numerical solutions of linear and nonlinear Volterra or Volterra-Fredholm integro-differential equations are increasing in popularity.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

Durmaz

Amirali

Amiraliyev

2022

J. Appl. Math. Comput.

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In this paper, a linear singularly perturbed Fredholm integro-differential initial value problem with integral condition is being considered. On a Shishkin-type mesh, a fitted finite difference approach is applied using a composite trapezoidal rule in both; in the integral part of equation and in the initial condition. The proposed technique acquires a uniform second-order convergence in respect to perturbation parameter. Further provided the numerical results to support the theoretical estimates.

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A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation

Cited by 9 publications

References 16 publications

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

A problem with impulse actions for nonlinear ODEs

An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

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