An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017
Cengizci, Süleyman
In thiswork, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.
International Journal of Differential Equations

Suggestions

 A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind Kaya, Ruşen; Taşeli, Hasan (2022-01-01) A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation -y″(x)+v(x)y(x)=λy(x),x∈(-∞,∞) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to x∈ [ - ℓ, ℓ] , where ℓ is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are co...
 The Sturm-Liouville operator on the space of functions with discontinuity conditions Uğur, Ömür (2006-03-01) The Sturm-Liouville differential operators defined on the space of discontinuous functions, where the moments of discontinuity of the functions are interior points of the interval and the discontinuity conditions are linear have been studied. Auxiliary results concerning the Green's formula, boundary value, and eigenvalue problems for impulsive differential equations are emphasized.
 A parallel sparse algorithm targeting arterial fluid mechanics computations Manguoğlu, Murat; Sameh, Ahmed H.; Tezduyar, Tayfun E. (2011-09-01) Iterative solution of large sparse nonsymmetric linear equation systems is one of the numerical challenges in arterial fluid-structure interaction computations. This is because the fluid mechanics parts of the fluid + structure block of the equation system that needs to be solved at every nonlinear iteration of each time step corresponds to incompressible flow, the computational domains include slender parts, and accurate wall shear stress calculations require boundary layer mesh refinement near the arteria...
 An eigenfunction expansion for the Schrodinger equation with arbitrary non-central potentials Taşeli, Hasan; Uğur, Ömür (2002-11-01) An eigenfunction expansion for the Schrodinger equation for a particle moving in an arbitrary non-central potential in the cylindrical polar coordinates is introduced, which reduces the partial differential equation to a system of coupled differential equations in the radial variable r. It is proved that such an orthogonal expansion of the wavefunction into the complete set of Chebyshev polynomials is uniformly convergent on any domain of (r, theta). As a benchmark application, the bound states calculations...
 A SUPG Formulation for Solving a Class of Singularly Perturbed Steady Problems in 2D Cengizci, Süleyman; Uğur, Ömür; Srinivasan, Natesan (2020-09-02) In this presentation, approximate solutions of singularly perturbed partial differential equations are examined. It is a well-known fact that the standard Galerkin finite element method (GFEM) experiences some instability problems in obtaining accurate approximations to the solution of convection-dominated equations. Therefore, in this work, the Streamline-Upwind/Petrov-Galerkin (SUPG) method is employed to overcome the instability issues for the numerical solution of these kinds of problems. Furthermore,...
Citation Formats
S. Cengizci, “An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations,” International Journal of Differential Equations, pp. 1–8, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/51638.