EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS

2006 ◽  
Vol 17 (06) ◽  
pp. 861-876 ◽  
Author(s):  
Ch. TSITOURAS
Keyword(s):  
Initial Value Problem ◽  
Second Order ◽  
Initial Value ◽  
Step Method ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
Phase Errors ◽  
Matlab Program ◽  
Algebraic Order

We present a new explicit hybrid two step method for the solution of second order initial value problem. It costs only nine function evaluations per step and attains eighth algebraic order so it is the cheapest in the literature. Its coefficients are chosen to reduce amplification and phase errors. Thus the method is well suited for facing problems with oscillatory solutions. After implementing a MATLAB program, we proceed with numerical tests that justify our effort.

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 383-398 ◽  
Author(s):  
I. TH. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Numerical Tests ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
High Phase

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

scholarly journals A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 713
Author(s):  
Higinio Ramos ◽  
Ridwanulahi Abdulganiy ◽  
Ruth Olowe ◽  
Samuel Jator
Keyword(s):  
Initial Value Problems ◽  
Second Order ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Direct Numerical Solution ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Third Derivative

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs

2018 ◽  
Vol 29 (01) ◽  
pp. 1850002 ◽  
Author(s):  
J. M. Franco ◽  
L. Rández
Keyword(s):  
Scientific Literature ◽  
Hybrid Methods ◽  
Computational Cost ◽  
High Dispersion ◽  
Minimal Cost ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Order Of Accuracy ◽  
Algebraic Order

The construction of new two-step hybrid (TSH) methods of explicit type with symmetric nodes and weights for the numerical integration of orbital and oscillatory second-order initial value problems (IVPs) is analyzed. These methods attain algebraic order eight with a computational cost of six or eight function evaluations per step (it is one of the lowest costs that we know in the literature) and they are optimal among the TSH methods in the sense that they reach a certain order of accuracy with minimal cost per step. The new TSH schemes also have high dispersion and dissipation orders (greater than 8) in order to be adapted to the solution of IVPs with oscillatory solutions. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order explicit TSH methods are more efficient than other standard TSH or Numerov-type methods proposed in the scientific literature.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Sign in / Sign up

Export Citation Format

Share Document