scholarly journals The Taylor series method of order $p$ and Adams-Bashforth method on time scales

2021 ◽  
Author(s):  
Svetlin Georgiev ◽  
Inci Erhan
Keyword(s):  
Time Scales ◽  
Taylor Series ◽  
Initial Value Problems ◽  
Arbitrary Order ◽  
Trapezoidal Rule ◽  
Dynamic Equations ◽  
Series Method ◽  
Initial Value ◽  
Nonlinear Dynamic Equations ◽  
Taylor Series Method

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2 step Adams-Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

scholarly journals Estimates in the Taylor series method for polynomial total systems of PDEs

2021 ◽  
Vol 17 (1) ◽  
pp. 27-39
Author(s):  
Levon K. Babadzanjanz ◽  
◽  
Irina Yu. Pototskaya ◽  
Yulia Yu. Pupysheva ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Taylor Series ◽  
Series Method ◽  
Initial Value ◽  
Taylor Coefficients ◽  
Recurrence Formulas ◽  
The Cauchy Problem ◽  
Taylor Series Method ◽  
Systems Of Pdes ◽  
Ode Systems

Many of total systems of PDEs can be reduced to the polynomial form. As was shown by various authors, one of the best methods for the numerical solution of the initial value problem for ODE systems is the Taylor Series Method (TSM). In the article, the authors consider the Cauchy problem for the total polynomial PDE system, obtain the recurrence formulas for Taylor coefficients, and then formulate and prove a theorem on the accuracy of its solutions by TSM.

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