Derivation of continuous equations for density of burglars

The discrete model for movement of burglars is given by the equation as: where R(x, t) is assumed to be strictly positive across the whole spatial domain for all t.We can express the relation in terms of discrete spatial laplacian . For this, in continuous limit, we re-express p(x, t) in terms of p(x, t) and R(x, t) R(X, t), p(x, t) p(X, t), where X=lx, l, is a grid spacing

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Introduction

The discrete model for movement of burglars is given by the equation as:            , ( , ) 1, 2, , R x t x t h x t R x t R x t        R x t ,          x t P x t G h R x t R x t 1, 1 , 2, ,              where ( , ) R x t is assumed to be strictly positive across the whole spatial domain for all t .We can express the relation in terms of discrete spatial laplacian . For this, in continuous limit, we re-express ( , ) x t  in terms of ( , ) X t  and ( , ) ( , ) R x t R X t  , ( , ) ( , ) P x t P X t  , where X lx  , l is a grid spacing. Since probability is independent of length or time scales, therefore we can write ( , ) ( , ) P a X t a P X t  . With this notion and assuming l is small, terms of the above relation can be re-written as 1 ( , ) ( , ) ( , ) ( , ) X x t h t h X t h X t h l l            1 ( , ) ( , ) ( , ) ( , ) X R x t R t R X t R X t l l    Maths 2018; 3(2): 408-411 Md.

Najmul Hoda Research Scholar, Department of Mathematics, T.M. Bhagalpur University, Bhagalpur, Bihar, India Correspondence Md. Najmul Hoda Research Scholar, Department of Mathematics, T.M. Bhagalpur University, Bhagalpur, Bihar, 1 ( 2, ) ( 2, ) ( 2 , ) ( 2 , ) X R x t R t R X l t R X l t l l        R X      X 2     X l R X l R X     1 ( 2, ) ( 2, ) ( 2 , ) ( 2 , ) X R x t R t R X l t R X l t l l        2 2 R X R X l     X    2 2 X l R X l R X       1 ( 1, ) ( 1, ) ( , ) ( , ) X x t t X l t X l t l l                    X 2 1 1 0                 X X 2 R X R X f X X X R X 2 2 4     1 ( 1, ) ( 1, ) ( , ) ( , ) X x t t X l t X l t l l           2     X  R X R X X     2 0 4 ( ) R X   Since ( , ) ( , ) P x t R x t h  therefore,  P x t R x t h ( , ) ( , )       X X 2 R X f l X l R X l R X                 ( , ) ( , ) 2 2 R X R X l X X P t R t h l l     X         X 2 X l R X l R X 2             1 1 ( , ) ( , ) P X t R X t h l l X 2 2 R X l R X X l        X 2 R X l R X 2 ( , ) ( , ) P X t R X t h   ,                 2 th u s 1 , 1 , P x t P X t     2 2 2                      X X        R X l R X R X l X l   4 R X l R X   2       1 , R X t h   2 4 2 2               X     R X l R X l R X X l   4 2 R X l R X         Thus the above relation is reduced to                     R X X l     X X R X t X t h X l t R X l t R X t , , , 2 , ,     R X l R X   2        2 2       X R X t ,        R X R X l X l   X 2        R X l R X   2 X l t R X t h R X l t R X t , 1 , 2 , ,                   G h X 2 2 R X l R X X l        X 2 R X l R X 2                2 2 2 2         X X       R X l R X R X l X l   4  For diffusion of the Risk of victimization and density of burglars to spatial location The relation can be written as R X l R X   2       2   4 2 2       ( , )           X            R X l R X l R X X l   4 R X X t h X l R X l R X 2        R X l R X 2       R X              X l R X h R X l R X 1 2                   X X       R X f l X l R X l R X 2     G h 4 2 R X R X l     X     X 2      X l R X l R X 2       2 R X l R X R X R X l 2 4 2             X X    4        , 1 X t h f l R X h G h       (1)         X l R X l R X 2       2  8 2 2 R X R X l R X l R X           X ( ) w h ere ( ) ( ) ( 2 ) ( ) R X f l X l R X l R X        4        X l R X l R X 2       R X ( )  R X    X l R X l R X ( ) ( 2 ) ( )   X X     X l R X l R X 2         2 2 R X R X l     X    X 2      X l R X l R X 2       Here, (0 ) ( ) f X   2 2 R X R X l     X    X 2       X l R X l R X 2            X R X f l X l R X l R X      2     2 R X l R X R X R X l 2 4 2           X X        2 2 R X R X l 4     X R X 2 X l l R X             2 X l R X l R X 2 2       8 2 2 R X R X l R X l R X           X    4 X l R X l R X 2                       X    l X t h X t X t        X X      2 1 0 2 , , , 2       2 X X X R X R X f X X R X           2 , , R X t X t      X                2 3 2 R X t h G h R X t (1 , ,   X               X X X R X R X R X R X                  X t h X t X t R X t h , , , , 4 1 6 1 6              1 6 X R X     2 R X t X t l     X   X t   X X     2 , , , 2 , R X t   X                     2 2 1     X R X t X t l     X R X t h X t           X X     X X X 2 2 , , , , 2 , R X t   X                     2 4 R X R X X X R X     G h 2 R X R X     X     X  Dividing both sides by h and Taking limit as 0 h  , as 0 l  , 2 2 4 X R X     l 2l  D h 0 2 , and assuming  (constant), we then have 2 2 2               X X X R X R X R X R X R X   X t h X t          4 1 6 1 6    , , lim h  0 1 6 X R X h     2    X X h  0 lim ( , ) ( , ) ( , ) 2 l X t R X t X t h     X X            0            X X X X R X R X f X X X R X R X     2 ( , ) ( , ) R X t X t  X 2         2 2 R X t ( , )     X X X X             2 R X R X X X R X R X     2 R X t X t l     X                      X X 2 , , , , 2 , R X t X t G R X t   X   X     2 0         X X X         R X f X X R X     X t X t R X t , lim , ,       2      t h 0 2 2     X X X     2     2 R X t X t l     X R X R X X X R X R X         X X 2 , , , 2 , X t h R X t   X                  For simplicity, it can be written as 2 R X t X t l     X                      X X X X X X X X X R R R f 2 2 2 2 (0) 2 2 , , , , 2 , R X t X t G R X t   X R R R                         X t X t R X t X t          t X X 2 (0 ) [ ] X X X X R f 1 , , , D , 2 R       2 , , R X t X t      X By maclaurians’ series expansion, we have G R X t ,   X 2 l f l f lf f O l                             As we assumed that density of burglars diffuse to spatial 3 0 0 0 2 ! where 3 ( ) O l represents the terms containing higher power of l .

As l is small, neglecting the term 3 ( ) O l and substituting location, so that ( , ) ( ) X t X    , ( , ) ( ) R X t R X  , thus the equation reduces to     X    R X X X t X     t X X 2 1 , D 2 R X   X the value of (0 ) f , (0 ) f  and (0 ) f  in the above relation, we have                          X  X R X G     R X t X t l f l X t X t      X X 2 2 , , , , 2 , R X t   X   For simplicity the resulting equation can be written as                        (2) With the help of relation (2), the equation (1) can be re- written as R X X X R G t R 1 2 D 2 X                        1 X X X D 2 R D R G t R X                  (3) Note that    2 R R R R R R               ( ) X X          R R R X R X    X X X X X 2   R R R R R R         2    X X X X X 2 R R R R R R          R X R X         Also 2 n 2        X X         d iv X  2 1 i i where  is the discrete spatial laplacian. Using these notations, we can express the above equation (3) as   2 1 D 2 D R R G t R                  (4)

Conclusion

Thus the continuous equation of Risk of victimization field is given by the equation and the continuous equation of density of burglars is given by the equation (4).

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