An Effective Tool: Numerical Solutions by Legendre Polynomials for High-order Linear Complex Differential Equations

2015 ◽  
Vol 8 (4) ◽  
pp. 348-355 ◽  
Author(s):  
Faruk Düşünceli ◽  
Ercan Çelik
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
High Order ◽  
Linear Complex ◽  
Order Linear ◽  
Complex Differential Equations

scholarly journals A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Faezeh Toutounian ◽  
Emran Tohidi ◽  
Stanford Shateyi
Keyword(s):  
Differential Equations ◽  
Matrix Equation ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
High Order ◽  
Linear Complex ◽  
Order Linear ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Complex Differential Equations

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given.

scholarly journals On the Growth of Solutions of Second Order Linear Complex Differential Equations whose Coefficients Satisfy Certain Conditions

2020 ◽  
Vol 17 (2) ◽  
pp. 0530
Author(s):  
Ayad Alkhalidy ◽  
Eman Hussein
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Infinite Order ◽  
Second Order ◽  
The Other ◽  
Nontrivial Solutions ◽  
Linear Complex ◽  
Order Linear ◽  
Complex Differential Equations ◽  
Growth Of Solutions

In this paper, we study the growth of solutions of the second order linear complex differential equations  insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

scholarly journals On Conjugacy of High-Order Linear Ordinary Differential Equations

1994 ◽  
Vol 1 (1) ◽  
pp. 1-8
Author(s):  
T. Chanturia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
High Order ◽  
Summable Function ◽  
Order Linear ◽  
Linear Ordinary Differential Equations

Abstract It is shown that the differential equation u (n) = p(t)u, where n ≥ 2 and p : [a, b] → ℝ is a summable function, is not conjugate in the segment [a, b], if for some l ∈ {1, . . . , n – 1}, α ∈]a, b[ and β ∈]α, b[ the inequalities hold.

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