Schemes of Analysis for Systems with Different Forms of Aftereffect

Deterministic and Stochastic Models

Vladimir V.Malanin, Igor E.Poloskov

Perm State University, Perm, Russia

Talk Outline

The Main Idea

Deterministic Differential Equations with Divergent Arguments

★ Systems of ODEqs with multiple constant delays ★ Neutral DEqs ★ DEqs with variable delays ★ PDEqs with constant delays

Stochastic Differential Equations with Delays

★ Systems of SDEqs with multiple constant delays ★ Stochastic neutral differential equations ★ SDEqs with a single variable delay

Analysis of Stochastic Integro - Differential Equations

★ Reduction of SIDEqs to SDEqs. Approximate numeric schemes ★ Iterative scheme for computation of Green matrix - function

Resume

The Main Idea

Systems with Hereditary Effects

Automatic control for technical devices and engineering procedures

Development of economic and social systems

Combustion in liquid  jet engines

Neutron moderation

Effects of radiations

A radio - location, radar and radio - navigation

Autonomous vessel course stabilization

Oscillations in vacuum - tube generators

Struggle for existence in biology

etc.

Extention of the phase vector

Pic01=Import["d:\\__delay\\pic01.jpg"]

x(t), t ≥ (Subscript[t, 0]) -> (Subscript[x, 0])(s), s ∈ [0,τ], {(Subscript[x, 0])(s), (Subscript[x, 1])(s)}, s ∈ [0,τ], ...

Equations without delay

The same time segment

Ordinary numeric integrators

Deterministic Differential Equations with Divergent Arguments

Scheme of computations

To build a chain of phase vectors (Subscript[(OverBar[x]), 0]), (Subscript[(OverBar[x]), 1]), …, (Subscript[(OverBar[x]), N]), … with increasing dimensions, which satisfy sets of ordinary differential equations (ODEqs) without delays with taking into account of vectors’ behaviour on an initial segment

After changing of segment of argument varying, calculations are made 
(i) on [0, τ]  for

(OverBar[x])'(t)=(OverBar[f])((OverBar[x])(t),(OverBar[x])(t-τ),...,(OverBar[x])(t-kτ),t),τ>0,k≥ 1

and

(OverBar[x])'(t)=(OverBar[f])(x(t),(OverBar[x])(t-τ),...,(OverBar[x])(t-kτ),(OverBar[x])'(t-τ),...,(OverBar[x])'(t-kτ),t),τ> 0, k≥ 1;

(ii) on  [0,(Subscript[τ, 0]) ]  for (3) (a variable delay is approximated by a piecewise constant function)

x'(t)=f(x(t),x(t-τ(t))),  0≤τ≤ (Subscript[τ, 0])

A number of ODEqs under integration is increasing by addition of the dimension of the phase vector on each step

Systems of ODEqs with multiple constant delays

(OverBar[x])'(t)=(Subscript[(OverBar[f]), ν])((OverBar[x])(t),(OverBar[x])(t-τ),(OverBar[x])(t-2τ),...,(OverBar[x])(t-ντ),t),
(OverBar[x])((Subscript[t, 0]))=((OverBar[x])^( *0)),t>(Subscript[t, ν])=(Subscript[t, 0])+ντ,

(OverBar[x])'(t)=(Subscript[(OverBar[f]), 0])((OverBar[x])(t),t), t ∈ ((Subscript[t, 0]),(Subscript[t, 1])],

(OverBar[x])'(t)=(Subscript[(OverBar[f]), 1])((OverBar[x])(t),(OverBar[x])(t-τ),t),t ∈ ((Subscript[t, 1]),(Subscript[t, 2])],

.....................................

(OverBar[x])'(t)=(Subscript[(OverBar[f]), ν-1])((OverBar[x])(t),(OverBar[x])(t-τ),(OverBar[x])(t-2ν),...,(OverBar[x])(t-(ν-1)τ),t), t ∈ ((Subscript[t, ν-1]),(Subscript[t, ν])],

(Subscript[(OverBar[f]), q]) ( ∘,∘ ) = (({(Subscript[f, qi])(∘,∘)})^T), q = 1,2,...,ν-1

Notation:

s∈[0,τ],(Subscript[t, q])=(Subscript[t, 0])+q τ,q=0,1,2,...,(Subscript[(OverBar[x]), q])(s)=(OverBar[x])(s+(Subscript[t, q]))

(Subscript[Δ, q])=[(Subscript[t, q-1]),(Subscript[t, q])],(Subscript[(OverBar[z]), 0])=(Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[z]), 1])=col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1])),(Subscript[(OverBar[z]), 2])=col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1]),(Subscript[(OverBar[x]), 2])),...,

(Subscript[(OverBar[y]), q]) =(Subscript[(OverBar[x]), q])(0)=(Subscript[(OverBar[x]), q-1])(τ),col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1]),(Subscript[(OverBar[x]), 2]),...)=(({(Subscript[x, 01]),(Subscript[x, 02]),...,(Subscript[x, 0n]),(Subscript[x, 11]),(Subscript[x, 12]),...,(Subscript[x, 1n]),(Subscript[x, 21]),(Subscript[x, 22]),...,(Subscript[x, 2n]),...})^T)

On N - th step :

(Subscript[(OverBar[x]), 0])'(s)=f0((Subscript[(OverBar[x]), 0])(s),s+(Subscript[t, 0])),

(Subscript[(OverBar[x]), 1])'(s)=f1((Subscript[(OverBar[x]), 0])(s),(Subscript[(OverBar[x]), 1])(s),s+(Subscript[t, 1])),

....................

(Subscript[(OverBar[x]), ν-1])'(s)=(Subscript[(OverBar[f]), ν-1])((Subscript[(OverBar[x]), 0])(s),(Subscript[(OverBar[x]), 1])(s),...,(Subscript[(OverBar[x]), ν-1])(s),s+(Subscript[t, ν-1])),

(Subscript[(OverBar[x]), ν])'(s)=(Subscript[(OverBar[f]), ν])((Subscript[(OverBar[x]), 0])(s),(Subscript[(OverBar[x]), 1])(s),...,(Subscript[(OverBar[x]), ν])(s),s+(Subscript[t, ν])),

(Subscript[(OverBar[x]), ν+1])'(s)=(Subscript[(OverBar[f]), ν+1])((Subscript[(OverBar[x]), 1])(s),(Subscript[(OverBar[x]), 2])(s),...,(Subscript[(OverBar[x]), ν+1])(s),s+(Subscript[t, ν+1])),

.............................

(Subscript[(OverBar[x]), N-1])'(s)=(Subscript[(OverBar[f]), N-1])((Subscript[(OverBar[x]), N-ν+1])(s),(Subscript[(OverBar[x]), N-ν+2])(s),...,(Subscript[(OverBar[x]), N-1])(s),s+(Subscript[t, N-1])),

(Subscript[(OverBar[x]), N])'(s)=(Subscript[(OverBar[f]), N])((Subscript[(OverBar[x]), N-ν]),(Subscript[(OverBar[x]), N-ν+1])(s),...,(Subscript[(OverBar[x]), N])(s),s+(Subscript[t, N])),

...............................

A procedure call

AfterEffect[eqs_,vars_,tq_,tauq_,initVal_,t0_,tk_,delta_,stepsPr_,maxSt_,meth_]

A sense of parameters

★ eqs is a vector of sets of DEqs sets on segments in the straight order in the standard Mathematica sense
★ vars is a vector of unknowns
★ tq is a name of time variable
★ tauq is a name of delay variable
★ initVal is a vector of initial values (for the first set of DE)
★ t0 is an initial value of time
★ tk is a final value of time
★ delta is a value of delay
★ stepsPr is a value of print steps on a delay segment
★ maxSt is a maximum value of steps
★ meth is a name of method

Example. Equations of Lottka - Volterra

(Subscript[x, 1])'=α [1-(Subscript[x, 1])(t-τ)/m] (Subscript[x, 1])(t)-b (Subscript[x, 1])(t) (Subscript[x, 2])(t),

(Subscript[x, 2])'=-β  (Subscript[x, 2])(t)+c (Subscript[x, 1])(t-τ) (Subscript[x, 2])(t-τ)

<<"d:/progs/de_delay.m";Off[General::spell]

α=2;β=1.5; b=1;c=1;m=4;

eqs={
{x1'[s]==0,x2'[s]==0},
{x1'[s]==α (1-x1[s-h]/m)x1[s]-b x1[s] x2[s],
x2'[s]==-β x2[s]+c x1[s-h] x2[s-h]}
};
vars={x1,x2};

rez=AfterEffect[eqs,vars,s,h,{5,1},-1,80,1,10,Automatic,Automatic];

rez1=Transpose[Drop[Transpose[rez],1]];

ListPlot[rez1,PlotJoined->True, PlotRange->{{0,11},{0,3.5}},Frame->True,ImageSize->{390,260}]

Example. Equation of Minorsky (Duffing with delay)

x''(t)+2 α x'(t)+(ω^2)x(t)+2 β x'(t-τ)+γ  (x^3)(t-τ)=0

Example. Equation of Van der Pol - Duffing

x''(t)+2(Subscript[α, 1])[(x^2)(t) – 1] x'(t)+2(Subscript[α, 2])[(x^2)(t-τ) – 1] x'(t)+2(Subscript[α, 3])[(x^2)(t) – 1] x'(t-τ)+2 (Subscript[α, 4])[(x^2)(t-τ) – 1] x'(t-τ)+((Subscript[c, 1])^2) x(t)+((Subscript[c, 2])^2) x(t-τ)+(Subscript[d, 1])(x^3)(t)+(Subscript[d, 2])(x^3)(t-τ)=0

Example. Equations of B - cell response dynamics

(Subscript[u, 1])' (t)=(Subscript[c, 1]) – (Subscript[c, 2]) (Subscript[u, 1])(t) – (Subscript[c, 3]) (Subscript[u, 2])(t) (Subscript[u, 1])(t),
(Subscript[u, 2])'(t)=(Subscript[c, 4]) (Subscript[u, 2])(t) – (Subscript[c, 5]) (Subscript[u, 2])(t) (Subscript[u, 3])(t),
(Subscript[u, 3])'(t)=(Subscript[c, 6]) (Subscript[u, 1])(t-τ) (Subscript[u, 2])(t-τ)H(t-τ) – (Subscript[c, 7]) (Subscript[u, 3])(t) – (Subscript[c, 8]) (Subscript[u, 2])(t) (Subscript[u, 3])(t)
τ = 10, 15

Example. Equation of an damped vibrator

Example. Equation of Van der Pol under an aperiodic excitation

Neutral DEqs

(OverBar[x])'(t)=(OverBar[f])((OverBar[x])(t),(OverBar[x])(t-τ),(OverBar[x])(t-2τ),...,(OverBar[x])(t-ντ),(OverBar[x])'(t-τ),(OverBar[x])'(t-2τ),...,(OverBar[x])'(t-ντ),t),
(OverBar[x])((Subscript[t, 0]))=((OverBar[x])^( *0)),t>(Subscript[t, ν])=(Subscript[t, 0])+ντ,

(OverBar[x])'(t)=(Subscript[(OverBar[f]), 0])((OverBar[x])(t),t), t ∈ ((Subscript[t, 0]),(Subscript[t, 1])],

(OverBar[x])'(t)=(Subscript[(OverBar[f]), 1])((OverBar[x])(t),(OverBar[x])(t-τ),(OverBar[x])'(t-τ),t),t ∈ ((Subscript[t, 1]),(Subscript[t, 2])],

.....................................

(OverBar[x])'(t)=(Subscript[(OverBar[f]), ν-1])((OverBar[x])(t),(OverBar[x])(t-τ),(OverBar[x])(t-2ν),...,(OverBar[x])(t-(ν-1)τ),(OverBar[x])'(t-τ),(OverBar[x])'(t-2ν),...,(OverBar[x])'(t-(ν-1)τ),t), t ∈ ((Subscript[t, ν-1]),(Subscript[t, ν])],

A procedure call

NeitrDEq[eqs_,vars_,tq_,tauq_,initEqs_,initVal_,t0_,tk_,delta_,stepsPr_,maxSt_,meth_]

Example. Model equation

x'(t)+x'(t-τ)+x(t)+x(t-τ)=0

<<"d:/progs/neitrdeq.m";

τ=1/2;k=1/4;α=1/2;

eqs={u'[s]+ u'[s-h]+u[s-h]==-u[s]};
initEqs={u'[s]==0};vars={u};initVal={5};

rez=NeitrDEq[eqs,vars,s,h,initEqs,initVal,-τ,10τ,τ,10,300000,StiffnessSwitching];

ListPlot[rez,PlotJoined->True,PlotRange->All,ImageSize->{300,200},PlotStyle->Thick]

Equation of autooscillation in a long electric line

Example. Model equation

Example. Mechanical system

M y''(t) + C  y'(t) + K  y(t)+m y''(t-τ)+m  L [θ''(t-τ)sin θ(t-τ)+((θ')^2)(t)cos θ(t-τ)]=0,m (L^2)θ''(t)+k  θ'(t)+m  g  L sin θ(t)+m L  y''(t) sin θ(t)=0

DEqs with variable delays

Example. Equation of pantograph

x'(t)=a x(t)+b x(q t),a,b,q=const (0<q<1)

Example. Equation with variable delay

x'(t)=a  x(t)+b  x(t-τ(t)),   a,b=const,   τ(t)=(sin^2)t

LinVarDel[tau_,a_,b_,x0_,nint_,TT_,del0_,xx1_,xx2_]:=Module[{eqs,i,j,gr,init,init0,m,newv,oldv,rez,t1,t2,t3,t4,vars},
del=TT/nint;
t2=0;gr={};init0={x0}; rez={{0,x0}};
For[i=1,i≤nint,i++,t1=t2;t2+=del;
vars={};eqs={};init={};t4=0;
For[j=1,j≤i,j++, t3=t4;t4+=del;
m=Floor[(t3 +del0-tau[t3])/del];
newv=ToExpression[StringJoin["x",ToString[j],"[t]"]];
oldv=ToExpression[StringJoin["x",ToString[j-m],"[t]"]];
AppendTo[vars,newv];
AppendTo[init,(newv /. t->0)==init0[[j]]];
AppendTo[eqs, D[newv,t]==a newv+b oldv]
];
{sol}=NDSolve[Join[eqs,init],vars,{t,0,del}];
init0=Join[{x0},(vars/.sol/.t->del)];
AppendTo[rez,{t2,(vars[[i]]/.sol/.t->del)}]
];
Return[rez]
]

tauv[w_]:=Sin[w]^2

TT=10;a=-5/2;b=1;nint=100;x0=1;del0=10^-8;
rez=LinVarDel[tauv,a,b,x0,nint,TT,del0];

ListPlot[rez,PlotJoined->True,PlotRange->{0,1},PlotStyle->Thick]

PDEqs with constant delays

Example. Equation of Burgers with delay

Example. Equation of “burning " with delay

Stochastic Differential Equations with Delays

Systems with Multiple Constant Delays

(OverDot[(OverBar[x])])(t) = (Subscript[(OverBar[f]), ν])((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),(Subscript[(OverBar[x]), 2τ])(t),...,(Subscript[(OverBar[x]), ντ])(t),t)+ G((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),(Subscript[(OverBar[x]), 2τ])(t),...,(Subscript[(OverBar[x]), ντ])(t),t)(OverBar[ξ])(t),   t>(Subscript[t, ν])=(Subscript[t, 0])+ντ.

Here (OverBar[x]) ∈ (R^n) is the phase vector, ξ ∈ (R^m) a vector of independent Gaussian white noises (M[(OverBar[ξ])(t)] = 0, M[(OverBar[ξ])(t) ((OverBar[ξ])^T)(t')] = E δ(t - t')), τ is a constant delay, ν > 0 is an integer, (OverBar[f]) = { (Subscript[f, i]) }^T: (R^((ν+1)n))×[(Subscript[t, 0]), ∞) -> (R^n) and G = { (Subscript[g, ij]) } :  (R^((ν+1)n))×[(Subscript[t, 0]), ∞)-> (R^n)× (R^m) are deterministic vector- and matrix-function respectively, (Subscript[(OverBar[x]), qτ])  = (Subscript[(OverBar[x]), qτ])(t) = (OverBar[x])(t - q τ), T is a symbol of the transposition, M stands for the mathematical expectation, E  is the identity matrix.

We suppose that the phase vector (OverBar[x])(t) being characterized by the probability density function (PDF) p((OverBar[x]), t) on the intervals ((Subscript[t, 0]), (Subscript[t, 1])], ((Subscript[t, 1]), (Subscript[t, 2])], ..., ((Subscript[t, ν-1]), (Subscript[t, ν])] satisfies the following systems of stochastic differential equations (SDEqs)

(OverDot[(OverBar[x])])(t) = (Subscript[(OverBar[f]), 0])((OverBar[x])(t),t)+ G((OverBar[x])(t),t)(OverBar[ξ])(t),

(OverDot[(OverBar[x])])(t) = (Subscript[(OverBar[f]), 1])((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),t)+ G((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),t)(OverBar[ξ])(t),

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ....

(OverDot[(OverBar[x])])(t) = (Subscript[(OverBar[f]), ν-1])((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),(Subscript[(OverBar[x]), 2τ])(t),...,(Subscript[(OverBar[x]), (ν-1)τ])(t),t)+ G((OverBar[x])(t),(Subscript[(OverBar[x]), τ])(t),(Subscript[(OverBar[x]), 2τ])(t),...,(Subscript[(OverBar[x]), (ν-1)τ])(t),t)(OverBar[ξ])(t).

To study a random change of the vector (OverBar[x]) (t)  for t > (Subscript[t, 0]), we use our scheme for analysis of different SDEqs with delay which is based on the idea of a transformation of the non-Markov vector process to a Markovian one. For this purpose, we expand the phase space of the system and introduce the following notation :

s ∈ [0,τ],  (Subscript[t, q ])= (Subscript[t, 0]) + q τ,  q = 0, 1, 2, ...,  (Subscript[s, q ])= s + (Subscript[t, q]),

(Subscript[(OverBar[x]), q])(s) = (OverBar[x])((Subscript[s, q])),  (Subscript[(OverBar[ξ]), q])(s) = (OverBar[ξ])((Subscript[s, q])),

(Subscript[p, q])((Subscript[(OverBar[x]), q]),s) = p((Subscript[(OverBar[x]), q]),(Subscript[s, q])),  (Subscript[p, 0])((Subscript[(OverBar[x]), 0]),0) = ((OverBar[p])^0)((Subscript[(OverBar[x]), 0])),

(Subscript[Δ, q]) = [(Subscript[t, q-1]),(Subscript[t, q])],  (Subscript[(OverBar[z]), 0]) = (Subscript[(OverBar[x]), 0]),  (Subscript[(OverBar[z]), 1]) = col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1])),  (Subscript[(OverBar[z]), 2]) = col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1]),(Subscript[(OverBar[x]), 2])),  ...,

(Subscript[(OverBar[ξ]), q])(0) = (Subscript[(OverBar[ξ]), q-1])(τ),  (Subscript[(OverBar[y]), q]) = (Subscript[(OverBar[x]), q])(0) = (Subscript[(OverBar[x]), q-1])(τ),

(Subscript[p, q])((Subscript[(OverBar[x]), q]),0) = (Subscript[p, q-1])((Subscript[(OverBar[x]), q]),τ), col((Subscript[(OverBar[x]), 0]),(Subscript[(OverBar[x]), 1]),(Subscript[(OverBar[x]), 2]),...,(Subscript[(OverBar[x]), N])) = (({(Subscript[x, 01]),(Subscript[x, 02]),...,(Subscript[x, 0n]),(Subscript[x, 11]),(Subscript[x, 12]),...,(Subscript[x, 1n]),...,(Subscript[x, N1]),(Subscript[x, N2]),...,(Subscript[x, Nn]),})^T)  .

We construct a chain of Fokker - Planck - Kolmogorov - like Eqs

((∂(p^([N]))(((OverBar[z])^([N])),t))/(∂t))=(1/2) (∑_(i,j=1)^Nn)(((∂^2)[(Subscript[b, ij]^([N]))(((OverBar[z])^([N])),t) (p^([N]))(((OverBar[z])^([N])),t)])/(∂(Subscript[z, i]^([N])) ∂(Subscript[z, j]^([N])))) -
(∑_(i=1)^Nn)((∂[(Subscript[a, i]^([N]))(((OverBar[z])^([N])),t) (p^([N]))(((OverBar[z])^([N])),t)])/(∂(Subscript[z, i]^([N])))) ,

(Subscript[a, i]^([N])) = (Subscript[f, i]^([N]))+ (1/2) (∑_(j=1)^Nn)(∑_(k=1)^mN)((∂(Subscript[g, ik]^([N])))/(∂(Subscript[z, j]^([N]))))(Subscript[g, jk]^([N])),   (Subscript[b, ij]^([N])) = (∑_(k=1)^mN)(Subscript[g, ik]^([N]))(Subscript[g, jk]^([N]))

for the PDFs of the vectors (Subscript[(OverBar[z]), 0]),  (Subscript[(OverBar[z]), 1]), ...,  (Subscript[(OverBar[z]), N]), ... belonging to the family of embedded phase spaces (R^n) ⊂ (R^(2n)) ⊂ ... ⊂ (R^Nn) ⊂ ... and Eqs for moments of these vectors.

Example. A nonlinear system with multiple delays

Example. Motions of vehicles along rough roads

Example. An estimation of a river water quality

A water basin is modeled as a cascade of water reservoirs (multiple delays)

Stochastic Systems under Continuous and Discrete Fluctuations

(OverDot[x])(t) = α(t)x(t) + β(t) x(t) ξ(t) + γ(t) x(t-τ) + ν(t) x(t-τ) η(t) +  r(t),    t > (Subscript[t, 1]) = (Subscript[t, 0]) + τ,

(OverDot[x])(t) = (OverBar[α])(t)x(t) + (OverBar[β])(t) x(t) ξ(t) +  r(t),    (Subscript[t, 0]) < t ≤(Subscript[t, 1]),    x((Subscript[t, 0]))= (x^0) .

ξ(t) and  η(t) are independent white noises,r(t)  is  a Poisson noise,α(t),β(t),(OverBar[α])(t), (OverBar[β])(t),γ(t),ν(t) are known functions

ODEqs for the first moments were obtained on the base  of a chain of Kolmogorov-Feller Eqs. In the case of absence of delay, such equation has got the following form:

((∂p(x,t))/(∂t))=(1/2) (((∂^2)[(β^2)(x^2) p(x,t)])/(∂(x^2)))-((∂[(α+(β^2))x  p(x,t)])/(∂x))- μ(t) p(x,t) +  μ(t) (∫_(-∞)^∞)w(x-x',t) p(x',t) dx',    p(x,(Subscript[t, 0]))=(Subscript[p, 0])(x),

where μ(t) is an intensity of jumps, w(z, t) is a function describing a jump distribution,

(∫_(-∞)^(+∞))w(x-x')dx'= 1.

Stochastic Neutral Differential Equations

(OverDot[(OverBar[x])])(t) = P(t) (OverBar[x])(t)+Q(t)(OverBar[x])(t-τ)+R(t)(OverDot[(OverBar[x])])(t-τ)+(OverBar[c])(t)+H(t)(OverBar[ξ])(t),
t > (Subscript[t, 1]) = (Subscript[t, 0]) + τ,

(OverDot[(OverBar[x])])(t) = (Subscript[P, 0])(t) (OverBar[x])(t)+(Subscript[(OverBar[c]), 0])(t)+(Subscript[H, 0])(t)(OverBar[ξ])(t),   (Subscript[t, 0]) < t ≤(Subscript[t, 1]),  
(OverBar[x])((Subscript[t, 0]))= (Subscript[(OverBar[x]), 0])

(OverBar[x]) ∈ (R^n) is the phase vector; P(t), (Subscript[P, 0])(t), Q(t), R(t), H(t), (Subscript[H, 0])(t) and c(t), (Subscript[c, 0])(t) are known matrix- and vector-functions; τ = const > 0, (OverBar[ξ])(t) is a vector of independent white noises.

Equations satisfied by the mean values and the covariances were obtained.

Example.

(OverDot[x])(t)=p x(t)+q x(t-τ)+r (OverDot[x])(t-τ)+ h ξ(t),  t>0;
(OverDot[x])(t)=0, t≤ 0;
m(-τ)=(m^0); D(-τ)=(D^0).

p=-1; q=0.2; r=-1; h=0.2; τ=0.5; (Subscript[t, 0])=-τ; (m^0)=5; (D^0)=0.25

Example.

(Overscript[x, ..])(t)+(Subscript[c, 0]) (Overscript[x, ..])(t-τ)+ 2(Subscript[α, 1])(OverDot[x])(t)+ 2(Subscript[α, 2])(OverDot[x])(t-τ)+(Subscript[β, 1])x(t)+(Subscript[β, 2])x(t-τ)= γ ξ(t),

(Subscript[c, 0])=0.75; (Subscript[α, 1])=0.125; (Subscript[α, 2])=0.167; (Subscript[β, 1])=3; (Subscript[β, 2])=2; γ=0.1;
(m^0)={5,0}; (D^0)={{-0.25,0},{0,1/8}}; τ=0.5; (Subscript[t, 0])=-τ

Example.

Linear SDEqs with a Single Variable Delay

(OverDot[(OverBar[x])])(t) = P(t) (OverBar[x])(t)+Q(t)(OverBar[x])(t-τ)+(OverBar[c])(t)+H(t)(OverBar[ξ])(t),
t >  (Subscript[t, 0]) ,   (OverBar[x])((Subscript[t, 0]))= ((OverBar[x])^0),

(OverDot[(OverBar[y])])(t) = (Subscript[P, 0])(t) (OverBar[y])(t)+(Subscript[(OverBar[c]), 0])(t)+(Subscript[H, 0])(t)(OverBar[ξ])(t),   (Subscript[(OverBar[t]), 0]) < t <(Subscript[t, 0]),  
(OverBar[y])((Subscript[(OverBar[t]), 0]))= ((OverBar[y])^0),
(Subscript[(OverBar[t]), 0])≤min(t-τ),      t≥(Subscript[t, 0])

where (OverBar[x]) ∈ (R^n); P(t), Q(t), H(t), (Subscript[P, 0])(t), (Subscript[H, 0])(t) and c(t), (Subscript[c, 0])(t) are known matrix- and vector-functions; τ (t) ≥ 0; (OverBar[ξ])(t) is a vector of independent white noises.
The  scheme is based on a change of τ(t) by a piecewise function (OverBar[τ])(t), where (SuperStar[τ])  is a length of steps.

Linear SDEqs with a Single Variable Delay

(OverDot[(OverBar[x])])(t) = P(t) (OverBar[x])(t)+Q(t)(OverBar[x])(t-τ)+(OverBar[c])(t)+H(t)(OverBar[ξ])(t),
t >  (Subscript[t, 0]) ,   (OverBar[x])((Subscript[t, 0]))= ((OverBar[x])^0),

(OverDot[(OverBar[y])])(t) = (Subscript[P, 0])(t) (OverBar[y])(t)+(Subscript[(OverBar[c]), 0])(t)+(Subscript[H, 0])(t)(OverBar[ξ])(t),   (Subscript[(OverBar[t]), 0]) < t <(Subscript[t, 0]),  
(OverBar[y])((Subscript[(OverBar[t]), 0]))= ((OverBar[y])^0),
(Subscript[(OverBar[t]), 0])≤min(t-τ),      t≥(Subscript[t, 0])

where (OverBar[x]) ∈ (R^n); P(t), Q(t), H(t), (Subscript[P, 0])(t), (Subscript[H, 0])(t) and c(t), (Subscript[c, 0])(t) are known matrix- and vector-functions; τ (t) ≥ 0; (OverBar[ξ])(t) is a vector of independent white noises.
The  scheme is based on a change of τ(t) by a piecewise function (OverBar[τ])(t), where (SuperStar[τ])  is a length of steps.

Example. Pantograph

(OverDot[x])(t) = a x(t) + b x(q t) + c ξ(t),  0 < t ≤ T,

x(0) = (Subscript[(OverBar[x]), 0]), a, b, c, d = const,  0 < q < 1,  
τ = t - q t =(1-q) t,  t ≥ 0.

Analysis of Stochastic Integro - Differential Equations

Reduction of SIDEqs to SDEqs

Example.

Example.

Iterative scheme for computation of Green matrix - function

Resume

Uselessness of special algorithms for an ODE numerical integration and a storage of prehistory

Simplicity of algorithms

Quickness of implementation

Thank you for attention!