We develop a mathematical model for the disease which can be
transmitted via vector and through blood transfusion in host
population. The host population is structured by the chronological
age. We assume that the instantaneous death and infection rates
depend on the age. Applying semigroup theory  and so forth, we
investigate the existence of equilibria. We also discuss local
stability of steady states.

Helong Liu

Houbao Xu

Jingyuan Yu

Guangtian Zhu

Stability on coupling SIR epidemic model with vaccination

The plane strain problem of determining strain energy release
rate, crack energy, and crack-opening displacement (COD) for a
moving Griffith crack at the interface of two dissimilar
orthotropic half-planes is considered. The problem is reduced to a
pair of singular integral equations of second kind which have
finally been solved by using Jacobi polynomials. Graphical plots
of the strain energy release rate, crack energy, and crack-opening
displacement for the problem in different particular cases are presented.

S.  Mukherjee

S.  Das

Moving interfacial Griffith crack between bonded dissimilar media

We investigate the existence of a global attractor and its
upper semicontinuity for the infinite-dimensional lattice
dynamical system of a partly dissipative reaction diffusion
system in the Hilbert space <mml:math alttext="$l^{2}\times l^{2}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>&#215;</mml:mo><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>. Such a system
is similar to the discretized FitzHugh-Nagumo system in
neurobiology, which is an adequate justification for its study.

Ahmed Y. Abdallah

Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

A theoretical analysis of  the Saint-Venant
one-dimensional flow model is performed in order to define the
nature of its instability. Following the Brigg  criterion, the
investigation is carried out by examining the  branch points
singularities of dispersion relation in the complex <mml:math alttext="$\omega$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#969;</mml:mi></mml:math> and
<mml:math alttext="$k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> planes, where <mml:math alttext="$\omega$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#969;</mml:mi></mml:math> and <mml:math alttext="$k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> are the complex pulsation
and wave number of the disturbance, respectively. The nature of
the linearly unstable conditions of flow is shown to be of
convective type, independently of the Froude number value.
Starting from this result a linear spatial stability analysis of
the one-dimensional flow model is performed, in terms of time
asymptotic response to a pointwise time periodic disturbance.
The study reveals an influence of the disturbance frequency on
the perturbation spatial growth rate, which constitutes the
theoretical foundation of semiempirical criteria commonly
employed for predicting roll waves occurrence.

C.  Di Cristo

A.  Vacca

On the convective nature  of roll waves instability

We analyse a model for pricing derivative securities in the
presence of both transaction costs as well as the risk from a
volatile portfolio. The model is based on the Black-Scholes
parabolic PDE in which transaction costs are described following
the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the
variance of the synthesized portfolio. Transaction
costs as well as the volatile portfolio risk depend on the time
lag between two consecutive transactions. Minimizing their sum
yields the optimal length of the hedge interval. In this model,
prices of vanilla options can be computed from a solution to a
fully nonlinear parabolic equation in which a diffusion
coefficient representing volatility nonlinearly depends on the
solution itself giving rise to explaining the volatility smile
analytically. We derive a robust numerical scheme for solving the
governing equation and perform extensive numerical testing of the
model and compare the results to real option market data. Implied
risk and volatility are introduced and computed for large option
datasets. We discuss how they can be used in qualitative and
quantitative analysis of option market data.

Martin  Janda&#269;ka

Daniel  &#352;ev&#269;ovi&#269;

On the risk-adjusted pricing-methodology-based valuation
of vanilla options and explanation of the volatility smile

We prove existence and uniform stability of strong solutions to a
quasilinear wave equation with a locally distributed nonlinear dissipation with source term of power nonlinearity of
the type <mml:math alttext="$u''-M (\int_{\Omega}|\nabla u| ^{2}dx)\Delta u+a(x)g(u') + f(u)=0 ,$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>u</mml:mi><mml:mo>&#8243;</mml:mo></mml:msup><mml:mo>&#8722;</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mo>&#8747;</mml:mo><mml:mi>&#937;</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mo>&#8711;</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>&#916;</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math> in <mml:math alttext="$\Omega\times] 0, +\infty[$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#937;</mml:mi><mml:mo>&#215;</mml:mo><mml:mrow><mml:mo>]</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>[</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$u=0,$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math> on <mml:math alttext="$\Gamma\times]0, +\infty[, $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#915;</mml:mi><mml:mo>&#215;</mml:mo><mml:mrow><mml:mo>]</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>[</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math><mml:math alttext="$u(x, 0)=u_{0}(x),\ u'(x,0) =u_{1}(x)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, in <mml:math alttext="$\Omega $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#937;</mml:mi></mml:math>.

E. Cabanillas Lapa

Z. Huaringa Segura

F. Leon Barboza

On the global solvability of solutions to a quasilinear
wave equation with localized damping and source terms

We construct Jacobi-weighted orthogonal polynomials
<mml:math alttext="$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w), \ \alpha,\beta,\gamma &#62; -1, \ \alpha+\beta+\gamma=0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi><mml:mo>&#62;</mml:mo><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>&#945;</mml:mi><mml:mo>+</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>+</mml:mo><mml:mi>&#947;</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>, on the triangular domain <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>. We show that these polynomials <mml:math alttext="$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> over the triangular domain <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> satisfy the following properties: <mml:math alttext="$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)\in {\mathcal L}_{n}$, $n\geq 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8712;</mml:mo><mml:msub><mml:mi>&#8466;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>1</mml:mn></mml:math>, <mml:math alttext="$r=0,1,\dotsc,n,$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo></mml:math> and <mml:math alttext="${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w) \perp {\mathcal P}_{n,s}^{(\alpha,\beta,\gamma)}(u,v,w)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8869;</mml:mo><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> for <mml:math alttext="$r\neq s$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi><mml:mo>&#8800;</mml:mo><mml:mi>s</mml:mi></mml:math>. And hence, <mml:math alttext="${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#947;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$n=0,1,2,\ldots$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo></mml:math>, <mml:math alttext="$r=0,1,\dotsc,n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:math> form an orthogonal system over the triangular 
domain <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> with respect to the Jacobi weight function. These
Jacobi-weighted orthogonal polynomials on triangular domains are
given in Bernstein basis form and thus preserve many properties of
the Bernstein polynomial basis.

A.  Rababah

M.  Alqudah

Jacobi-weighted orthogonal polynomials on triangular domains

Steady incompressible flow around a circular cylinder in an
external magnetic field that is aligned with fluid flow direction
is studied for <mml:math alttext="$\mathrm{Re}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>Re</mml:mtext></mml:mrow></mml:math> (Reynolds number) up to 40 and the
interaction parameter in the range <mml:math alttext="$0\leq N \leq 15$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>N</mml:mi><mml:mo>&#8804;</mml:mo><mml:mn>15</mml:mn></mml:math> (or <mml:math alttext="$0\leq M \leq 30$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>M</mml:mi><mml:mo>&#8804;</mml:mo><mml:mn>30</mml:mn></mml:math>), where <mml:math alttext="$M$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math> is the Hartmann number related to <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> by the
relation <mml:math alttext="$M = \sqrt{2 N \mathrm{Re}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>N</mml:mi><mml:mtext>Re</mml:mtext></mml:mrow></mml:msqrt></mml:math>, using finite difference method.
The pressure-Poisson equation is solved to find pressure fields in
the flow region. The multigrid method with defect correction
technique is used to achieve the second-order accurate solution of
complete nonlinear Navier-Stokes equations. It is found that the
boundary layer separation at rear stagnation point for <mml:math alttext="$\mathrm{Re} = 10$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:math>
is suppressed completely when <mml:math alttext="$N \lt 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&#60;</mml:mo><mml:mn>1</mml:mn></mml:math> and it started growing again
when <mml:math alttext="$N\geq 9$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>9</mml:mn></mml:math>. For <mml:math alttext="$\mathrm{Re} = 20$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>20</mml:mn></mml:math> and 40, the suppression is not
complete and in addition to that the rear separation bubble
started increasing when <mml:math alttext="$N\geq 3$s" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>3</mml:mn></mml:math>. The drag coefficient decreases
for low values of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math><mml:math alttext="$(\lt 0.1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#60;</mml:mo><mml:mn>0.1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and then increases with increase
of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>. The pressure drag coefficient, total drag coefficient, and
pressure at rear stagnation point vary with <mml:math alttext="$\sqrt{N}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mi>N</mml:mi></mml:msqrt></mml:math>. It is also found that the upstream and downstream pressures on the surface of
the cylinder increase for low values of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math><mml:math alttext="$(\lt 0.1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#60;</mml:mo><mml:mn>0.1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and rear pressure inversion occurs with further increase of <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>. These results are in agreement with experimental findings.

T. V. S. Sekhar

R.  Sivakumar

T. V. R. Ravi Kumar

Drag and pressure fields for the MHD flow around a circular cylinder at intermediate Reynolds numbers

We study the numerical solvability of a class of nonlinear weakly singular integral equations of Volterra-Hammerstein type with noncompact kernels. We obtain existence and uniqueness results and analyze the product integration methods for these equations under some verifiable conditions on the kernels and nonlinear functions. The convergence analysis is investigated and finally numerical experiments are given, which confirm our theoretical results.

M.  Hadizadeh

M.  Mohamadsohi

Numerical solvability of a class of Volterra-Hammerstein integral equations with noncompact kernels

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio-dependent predator-prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.

Fengde Chen

Periodicity in a ratio-dependent predator-prey system with stage structure for predator

We prove existence results for solutions of periodic boundary value problems concerning the <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>th-order differential equation
with <mml:math alttext="$p$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-Laplacian <mml:math alttext="$[\phi (x^{(n-1)}(t))]'  =  f(t, x(t), x'(t),  \dotsc,\linebreak x^{(n-1)}(t))$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>&#966;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>'</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>'</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and the boundary value conditions <mml:math alttext="$x^{(i)}(0)\!=\! x^{(i)}(T)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$i= 0, \dotsc, n-1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:math>. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> may be a polynomial and the degree of some variables among <mml:math alttext="$x_0, x_1, \dotsc, x_{n-1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> in the function <mml:math alttext="$f(t, x_0, x_1, \dotsc, x_{n-1})$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> is allowed to be greater than <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>.

Yuji  Liu

Weigao  Ge

Periodic boundary value problems for <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g></svg>th-order ordinary differential equations with <svg style="vertical-align:-3.50804pt" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-laplacian

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<p><em>Journal of Applied Mathematics</em> publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.</p>


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<p>Chief Editor, Professor Theodore E. Simos, is based at Ulyanovsk State Technical University, Russia. His main research interest is the numerical analysis of differential equations.</p>


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Journal of Applied Mathematics publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.

Journal of Applied Mathematics