scholarly journals Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Wei Nian Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Forced Oscillation ◽  
Smooth Boundary ◽  
Sufficient Conditions ◽  
Normal Vector ◽  
Fractional Partial Differential Equations ◽  
Damping Term ◽  
Partial Differential ◽  
Liouville Fractional Derivative

Sufficient conditions are established for the forced oscillation of fractional partial differential equations with damping term of the form(∂/∂t)(D+,tαu(x,t))+p(t)D+,tαu(x,t)=a(t)Δu(x,t)-q(x,t)u(x,t)+f(x,t),(x,t)∈Ω×R+≡G, with one of the two following boundary conditions:∂u(x,t)/∂N=ψ(x,t),  (x,t)∈∂Ω×R+oru(x,t)=0,  (x,t)∈∂Ω×R+, whereΩis a bounded domain inRnwith a piecewise smooth boundary,∂Ω,R+=[0,∞),  α∈(0,1)is a constant,D+,tαu(x,t)is the Riemann-Liouville fractional derivative of orderαofuwith respect tot,Δis the Laplacian inRn,Nis the unit exterior normal vector to∂Ω, andψ(x,t)is a continuous function on∂Ω×R+. The main results are illustrated by some examples.

scholarly journals Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shaochun Ji ◽  
Dandan Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Integrodifferential System ◽  
Liouville Fractional Derivative

AbstractThis paper is concerned with the semilinear fractional integrodifferential system with Riemann–Liouville fractional derivative. Firstly, we introduce the suitable $C_{1-\alpha }$C1−α-solution to Riemann–Liouville fractional integrodifferential equations in the new frame of fractional resolvents. Some properties of fractional resolvents are given. Then we discuss the sufficient conditions for the existence of solutions without the Lipschitz assumptions to nonlinear item. Finally, an example on fractional partial differential equations is presented to illustrate our abstract results.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

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