scholarly journals Cauchy Problems for Evolutionary Pseudodifferential Equations overp-Adic Field

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Bo Wu ◽  
Yin Li ◽  
Weiyi Su
Keyword(s):  
Exact Solutions ◽  
Pseudodifferential Operator ◽  
Second Order ◽  
Cauchy Problems ◽  
Pseudodifferential Equations ◽  
Mixed Classes

We study a class of evolutionary pseudodifferential equations of the second order int,  (∂2u(t,x)/∂t2+2a2Tα/2(∂u(t,x)/∂t)+b2Tαu(t,x)+c2u(t,x)=q(t,x)), wheret∈(0,z]andTαis pseudodifferential operator inx∈Qp, which defined by Weiyi Su in 1992. We obtained the exact solutions to the equations which belong to mixed classes of real andp-adic functions.

scholarly journals A Partial Lagrangian Approach to Mathematical Models of Epidemiology

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
I. Naeem ◽  
F. M. Mahomed
Keyword(s):  
Mathematical Models ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Order Equation ◽  
Second Order ◽  
Lagrangian Approach ◽  
Hiv Model ◽  
First Order ◽  
Partial Lagrangian

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

Lp‐maximal regularity for incomplete second‐order Cauchy problems

Kybernetes ◽  
2010 ◽  
Vol 39 (6) ◽  
pp. 954-960
Author(s):  
Yongzhong Huang ◽  
Yan Feng
Keyword(s):  
Maximal Regularity ◽  
Second Order ◽  
Cauchy Problems

scholarly journals On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients

2008 ◽  
Vol 102 (2) ◽  
pp. 283 ◽  
Author(s):  
Massimo Cicognani ◽  
Fumihiko Hirosawa
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Second Order ◽  
Time Interval ◽  
Well Posedness ◽  
Cauchy Problems ◽  
End Point ◽  
Stabilization Condition ◽  
Loss Of Regularity

We consider the loss of regularity of the solution to the backward Cauchy problem for a second order strictly hyperbolic equation on the time interval $[0,T]$ with time depending coefficients which have a singularity only at the end point $t=0$. The main purpose of this paper is to show that the loss of regularity of the solution on the Gevrey scale can be described by the order of differentiability of the coefficients on $(0,T]$, the order of singularities of each derivatives as $t\to0$ and a stabilization condition of the amplitude of oscillations described by an integral on $(0,T)$. Moreover, we prove the optimality of the conditions for $C^\infty$ coefficients on $(0,T]$ by constructing a counterexample.

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