scholarly journals Solving Delay Differential Equations by an Accurate Method with Interpolation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approximation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Reproducing Kernel Method ◽  
Comparison Of The Results ◽  
Delay Differential

We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

scholarly journals Numerical Solution of Delay Differential Equations via the Reproducing Kernel Hilbert Spaces Method

2020 ◽  
pp. 1-14
Author(s):  
Reham K. Alshehri ◽  
Banan S. Maayah ◽  
Abdelhalim Ebaid
Keyword(s):  
Differential Equations ◽  
Hilbert Spaces ◽  
Delay Differential Equations ◽  
Reproducing Kernel ◽  
Approximate Solutions ◽  
Reproducing Kernel Hilbert Spaces ◽  
Analytical Technique ◽  
Kernel Hilbert Spaces ◽  
Past Data ◽  
Delay Differential

Delay differential equations (DDEs) are generalization of the ordinary differential equation (ODEs), which is suitable for physical system that also depends on the past data. In this paper, the Reproducing Kernel Hilbert Spaces (RKHS) method is applied to approximate the solution of a general form of first, second and third order fractional DDEs (FDDEs). It is a relatively new analytical technique. The analytical and approximate solutions are represented in terms of series in the RKHS.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

2019 ◽  
Vol 30 (11) ◽  
pp. 4711-4733 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiphase Flows ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Fractional Partial Differential Equations ◽  
Reproducing Kernel Method ◽  
Content Type

Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

scholarly journals A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
F. Z. Geng ◽  
X. M. Li
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Initial Position ◽  
New Method ◽  
Riccati Differential Equation ◽  
Reproducing Kernel Method ◽  
Riccati Differential Equations ◽  
Linear Problems ◽  
Quasilinearization Technique

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

scholarly journals Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
D. Olvera ◽  
A. Elías-Zúñiga ◽  
L. N. López de Lacalle ◽  
C. A. Rodríguez
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Numerical Integration ◽  
Delay Differential Equations ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Integration Algorithm ◽  
Homotopy Perturbation ◽  
Delay Differential

We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.

BIFURCATIONS IN APPROXIMATE SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

2004 ◽  
Vol 14 (09) ◽  
pp. 2999-3021 ◽  
Author(s):  
CHRISTOPHER T. H. BAKER ◽  
JUDITH M. FORD ◽  
NEVILLE J. FORD
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Lag ◽  
Approximate Solutions ◽  
Prima Facie ◽  
Standard Wiener Process ◽  
Qualitative Behavior ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Differential

We consider stochastic delay differential equations of the form [Formula: see text] interpreted in the Itô sense, with Y(t)=Φ(t) for t∈[t0-τ,t0] (here, W(t) is a standard Wiener process and τ>0 is the constant "lag", or "time-lag"). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximate) solutions of the equation. Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler–Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field.

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