Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations via Cubic Spline in Tension

2016 ◽  
Vol 3 (3) ◽  
pp. 1703-1717 ◽  
Author(s):  
P. Pramod Chakravarthy ◽  
S. Dinesh Kumar ◽  
R. Nageshwar Rao
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Delay Differential Equations ◽  
Cubic Spline ◽  
Second Order ◽  
Singularly Perturbed ◽  
Delay Differential

scholarly journals SOLVING BOUNDARY VALUE PROBLEMS FOR SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS BY Ε-APPROXIMATE FIXED-POINT METHOD

2015 ◽  
Vol 20 (3) ◽  
pp. 369-381 ◽  
Author(s):  
Zbigniew Bartoszewski ◽  
Anna Baranowska
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Delay Differential Equations ◽  
Boundary Value ◽  
Cubic Spline ◽  
Second Order ◽  
Singularly Perturbed ◽  
Grid Points ◽  
Delay Differential

In this paper, the boundary value problem for second order singularly perturbed delay differential equation is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by cubic spline v h defined by its values v i = v h (t i ) at grid points t i , i = 0, 1, . . . , N. The necessary for construction the cubic spline and missing the first derivatives at the boundary are replaced by the derivatives of the corresponding interpolating polynomials matching the grid points values nearest to the boundary points. An approximation of the solution is obtained by minimization techniques applied to a function whose arguments are the grid point values of the sought spline. The results of numerical experiments with two boundary value problems for the second order singularly perturbed delay differential equations as well as their comparison with the results of other methods employed by other authors are also provided.

Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline

2018 ◽  
Vol 19 (3-4) ◽  
pp. 357-365 ◽  
Author(s):  
A. S. V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Cubic Spline ◽  
Singularly Perturbed ◽  
Delay Term ◽  
Second Order Convergence ◽  
Delay Differential ◽  
Parametric Cubic Spline

AbstractIn this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

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