scholarly journals Some Properties of a Solution to a Family of Inhomogeneous Linear Ordinary Differential Equations

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Bernstein Function ◽  
Linear Ordinary Differential Equations ◽  
The Family ◽  
Absolute Monotonicity ◽  
Significant Expression ◽  
Lerch Transcendent ◽  
Derivatives Of

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

scholarly journals A diagonal recurrence relation for the Stirling numbers of the first kind

2018 ◽  
Vol 12 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recurrence Relation ◽  
Stirling Numbers ◽  
Bernstein Function ◽  
Linear Ordinary Differential Equations ◽  
The Family ◽  
Significant Expression ◽  
Lerch Transcendent ◽  
Derivatives Of

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

2016 ◽  
Vol 7 (1) ◽  
pp. 19 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Operational Matrix ◽  
Variable Coefficients ◽  
High Order ◽  
Algebraic Equations ◽  
Linear Ordinary Differential Equations ◽  
The Given ◽  
Derivatives Of

The purpose of this paper is to investigate a new exponential Chebyshev (EC) operational matrix of derivatives. The new operational matrix of derivatives of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. This method transforms the given differential equation and conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of high-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.

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