An operator-difference scheme for abstract Cauchy problems

Ashyralyev, Allaberen; Köksal, Mehmet Emir; Agarwal, Ravi P.

doi:10.1016/j.camwa.2011.02.014

Cited by 12 publications

(7 citation statements)

References 35 publications

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“…x r x r (6) acting in the space of grid functions φ h (x) and satisfying the conditions φ h (x) = 0 for all x ∈ S h . It is known that A x h is a self-adjoint positive definite operator in L 2 ( Ω h ).…”

Section: Theorem 1 For the Solution Of Difference Scheme (4) The Fomentioning

confidence: 99%

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A second order of accuracy finite difference scheme for the integral-differential equation of the hyperbolic type

Direk¹,

Ashyraliyev²

2014

AIP Conference Proceedings

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In this paper, the second order of accuracy difference scheme approximately solving the initial-value problem for the integral-differential equation of the hyperbolic type in a Hilbert space H is presented. The stability estimates for the solution of this difference scheme are established. In application, the second order of accuracy difference scheme for solving the local boundary problem for the multidimensional integral-differential equation of the hyperbolic type with two dependent limits is constructed. Theoretical results are supported by numerical examples.

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Section: Theorem 1 For the Solution Of Difference Scheme (4) The Fomentioning

confidence: 99%

“…[1][2][3]. There is a great deal of work in constructing and analyzing difference schemes for numerical solutions of hyperbolic differential equations [4][5][6][7][8][9][10][11]. In [12], integral-differential equations of the hyperbolic type with two dependent limits have been studied.…”

Section: Introductionmentioning

confidence: 99%

A second order of accuracy finite difference scheme for the integral-differential equation of the hyperbolic type

Direk¹,

Ashyraliyev²

2014

AIP Conference Proceedings

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“…Moreover, it is known that various initial boundary value problems for ordinary and partial differential equations can be reduced to IVPs (see [9,16]). This reduction allows us to study differential equations with operator coefficient in abstract spaces (see [1,2,15,26] and the references therein). One of the most significant and important central point for a nonlinear IVP is to understand how nonlinearity affects the nature and characteristic of the solution.…”

Section: Introductionmentioning

confidence: 99%

Quasilinearization method for nonlinear differential equations with causal operators

Yakar¹,

Köksal²

2016

J. Nonlinear Sci. Appl.

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Employing quasilinearization technique coupled with the method of upper and lower solutions, we construct monotone sequences whose iterates are solutions to corresponding linear problems and show that the sequences converge uniformly and monotonically to the unique solution of the nonlinear problem with causal operator. Especially, instead of assuming convexity or concavity assumption on the nonlinear term that is demanded by the method of quasilinearization, we impose weaker conditions to be more useful in applications. The results obtained include several special cases and extend previous results.

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“…Dehghan and Shokri (2009) solved the one-and two-dimensional hyperbolic equations using collocation points and used the thin plate-spline radial basis functions to approximate the solution. Dehghan and Mohebbi (Dehghan and Mohebbi, 2008;Dehghan, 2006) (2011), Ashyralyev and Koksal (2008), Ashyralyev et al (2011), Behrouz et al (2013), Dehghan and Nikpour (2013a), Dehghan and Salehi (2012), Dehghan and Ghesmati (2010a, b), Dehghan and Shokri (2008), Manzari and Manzari (1998), Raftari et al (2013) and Dai et al, (2006). In this paper, the authors have developed a numerical algorithm based on cosine expansion-based differential quadrature method (CDQM) for the numerical solutions of two-space dimensional hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions.…”

mentioning

confidence: 97%

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

Verma

Jiwari

2015

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

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An operator-difference scheme for abstract Cauchy problems

Abstract: true2016-03-16T23:02:09

Cited by 12 publications

References 35 publications

A second order of accuracy finite difference scheme for the integral-differential equation of the hyperbolic type

A second order of accuracy finite difference scheme for the integral-differential equation of the hyperbolic type

Quasilinearization method for nonlinear differential equations with causal operators

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

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