A System Associated with the Confluent Hypergeometric Function Φ3 and a Certain Linear Ordinary Differential Equation with Two Irregular Singular Points

1997 ◽  
Vol 08 (05) ◽  
pp. 689-702 ◽  
Author(s):  
Shun Shimomura
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Linear Ordinary Differential Equation ◽  
Singular Points ◽  
Third Order ◽  
Regular Type ◽  
Singular Loci ◽  
Irregular Singular Points

The confluent hypergeometric function Φ3 satisfies a system of partial differential equations on P1(C) × P1(C) with the singular loci x = 0, x = ∞, y = ∞ of irregular type and y = 0 of regular type. We obtain asymptotic expansions and Stokes multipliers of linearly independent solutions near the singular loci x = 0 and x = ∞. Applying the results we also clarify the global behaviour of the solutions of a third order linear ordinary differential equation with two irregular singular points.

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

2020 ◽  
Vol 27 (4) ◽  
pp. 593-603 ◽  
Author(s):  
Kemal Özen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Adjoint Problem ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Order Linear ◽  
Theoretical Results

AbstractIn this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function Φ2, I

1993 ◽  
Vol 123 (6) ◽  
pp. 1165-1177
Author(s):  
Shun Shimomura
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function ◽  
Connection Formula ◽  
Regular Type ◽  
Linearly Independent ◽  
Singular Loci

SynopsisThe confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.

scholarly journals Classroom Note: Numerical solutions of Nagumo's equation

1999 ◽  
Vol 3 (2) ◽  
pp. 189-193 ◽  
Author(s):  
M. Iqbal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Solutions ◽  
Linear Ordinary Differential Equation ◽  
Bounded Solutions ◽  
Third Order ◽  
Non Linear

Nagumo's equation is a third order non-linear ordinary differential equation d3udx3−cd2udx2+f′(u)dudx−(b/c)u=0 where f(u)=u(1−u)(u−α) , 0<α<1 . In this paper we have developed a technique to determine those values of the parameters a,b and c which permit non-constant bounded solutions.

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