FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS

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1 UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS by Anna Baranowska Zdzis law Kamont Abstract. Classical solutions of nonlinear partial differential equations are approximated in te paper by solutions of quasilinear systems of difference equations. Sufficient conditions for te convergence of te metod are given. Te proof of te stability of te difference problem is based on a comparison metod. Tis new approac to te numerical solving of nonlinear equations is generated by a linearization metod for initial problems. Numerical examples are given.. Difference systems corresponding to nonlinear equations. For any metric spaces X Y we denote by CX, Y ) te class of all continuous functions from X into Y. We will use vectorial inequalities wit te understing tat te same inequalities old between teir corresponding components. Let E be te Haar pyramid E = { t, x) = t, x,..., x n ) R +n : t [0, a], b + Mt x b Mt } were a > 0, M = M,..., M n ) R n +, R + = [0, + ), b = b,..., b n ) R n b Ma. Write Ω = E R R n suppose tat f : Ω R is a given function of te variables t, x, p, q), q = q,..., q n ). We consider te nonlinear first order partial differential equation ) t zt, x) = f t, x, zt, x), x zt, x) ) wit te initial condition 2) z0, x) = ϕx), x [ b, b], were ϕ: [ b, b] R is a given function x z = x z,..., xn z). We are interested in te construction of a metod for te approximation of solutions to problem ), 2) wit solutions of associated difference equations in te

2 6 estimation of te difference between tese solutions. Te classical difference metods for nonlinear partial differential equations consist in replacing partial derivatives wit difference expressions. Ten, under suitable assumptions on given functions on te mes, solutions of difference equations approximate solutions of te original problem. Let N Z be te sets of natural numbers integers, respectively. For x, y R n, x = x,..., x n ), y = y,..., y n ), we write x y = x y,..., x n y n ) R n, x = x i. We define a mes on te set E in te following way. Suppose tat 0, ) were =,..., n ) st for steps of te mes. For = 0, ) i, m) Z +n were m = m,..., m n ), we define nodal points as follows: i= t i) = i 0, x m) = m, x m) = x m ),..., x mn) n ). Denote by te set of all = 0, ) suc tat tere is N = N,..., N n ) N n wit te property N = b. We assume tat tat tere is a sequence { j) }, j), suc tat lim j j) = 0. Tere is N 0 N suc tat N 0 0 a < N 0 + ) 0. Let R +n = { t i), x m) ) : i, m) Z +n } E = E R +n, E = { ti), x m) ) E : t i) + 0, x m) ) E }, E 0. = { x m) : N m N }, I = { t i) : 0 i N 0 }. For a function z : E R for a point t i), x m) ) E we write z i,m) = zt i), x m) ). For j n we put e j = 0,..., 0,, 0,..., 0) R n, sting on te j-t place. We define difference operators δ 0, δ = δ,..., δ n ) in te following way. For z : E R we put 3) δ 0 z i,m) = 0 z i+,m) z i,m) ), 4) δ j z i,m) = j z i,m+e j) z i,m) ), j κ, 5) δ j z i,m) = j z i,m) z i,m e j) ), κ + j n, were 0 κ n is fixed. If κ = 0 ten δ is given by 5), for κ = n, δ is defined by 4). Write δz i,m) = δ z i,m),..., δ n z i,m) ).

3 7 Suppose tat problem ), 2) is solved numerically by te difference metod 6) δ 0 z i,m) = f t i), x m), z i,m), δz i,m) ), 7) z 0,m) = ϕ m), xm) E 0., were ϕ : E 0. R is a given function. If we assume tat 0 M ten te set E as te following property: if t i), x m) ) E ten t i), x m+ej) ), t i), x m ej) ) E for j n consequently, tere exists exactly one solution z : E R of problem 6), 7). Sufficient conditions for te convergence of te metod 6), 7) to a solution of ), 2) are given in te following teorem. Teorem.. Suppose tat ) f CΩ, R) te derivatives q f,..., qn f ) = q f exist on Ω q f CΩ, R n ), 2) tere is A R + suc tat 8) ft, x, p, q) ft, x, p, q) A p p on Ω, 3), 0 M 0 were P = t, x, p, q) j qj f P ) 0 9) qj f P ) 0 for j κ, qj f P ) 0 for κ + j n, 4) v : E R is a solution of ), 2), v is of class C tere is a function α 0 : R + suc tat ϕ m) ϕ m) α 0 ) on E 0. lim 0 α 0 ) = 0, 5) z : E R is a solution of 6), 7). Under tese assumptions tere is a function α: R + suc tat 0) v i,m) z i,m) α) on E lim 0 α) = 0. Te above teorem is a consequence of results presented in [] [3], see also [4]. Note tat te Lipscitz condition 8) may be replaced in te teorem by a nonlinear estimate of te Perron type. Te following condition is important in tese considerations. Write ) sign q f = sign q f,..., sign qn f ). We ave assumed in Teorem. tat function ) is constant on Ω.

4 8 Remark.2. Suppose tat all te assumptions of Teorem. are satisfied ) te solution v : E R of ), 2) is of class C 2 c R + is suc a constant tat xj vt, x) c, tt vt, x) c, xj x j vt, x) c, j n, were t, x) E, 2) tere is A 0 R + suc tat q ft, x, p, q) A 0 on Ω. Ten we ave te following error estimate for te metod 6), 7): 2) v i,m) z i,m) ᾱ) on E were ᾱ) = α 0 )e Aa c A 0 M ) θa), M = max {M i : i n }, θa) = eaa if A > 0, θa) = a if A = 0. A Te above result can be proved by te metods used in [] [2]. Consider now anoter difference metod for problem ), 2). Let te operators δ 0, δ = δ,..., δ n ) be defined by 3) δ 0 z i,m) = 0 z i+,m) Dz i,m)), Dz i,m) = 2n z i,m+e j) + z i,m e j) ), 4) δ j z i,m) = 2 j z i,m+e j) z i,m e j) ), j n, were t i), x m) ) E z : E R. Consider difference problem 6), 7) wit δ 0 δ given by 3), 4). Teorem.3. Suppose tat conditions ), 2) of Teorem. are satisfied ), 0 M for P = t, x, p, q) Ω we ave n 0 j qj f P ) 0, j n, 2) v : E R is a solution of ), 2), v is of class C tere is a function α 0 : R + suc tat ϕ m) ϕ m) α 0) on E 0. lim 0 α 0 ) = 0, 3) z : E R is a solution of 6), 7) wit δ 0, δ given by 3), 4).

5 9 Ten tere is a function α: R + suc tat condition 0) is satisfied. Tis teorem can be proved wit use of te metods presented in [], [4]. Remark.4. Suppose tat all te assumtions of Teoren.3 are satisfied te solution v : E of ), 2) is of class C 2. Ten we ave te following error estimate for te metod: tere are c 0, c R + suc tat v i,m) z i,m) c 0 α 0 ) + c 0 on E. Now we formulate a new class of difference problems corresponding to ), 2). We transform te nonlinear differential equation into a quasilinear system of difference equations. We will use a linearization metod for equation ) wit respect to te last variable. We omit te condition tat function ) is constant on Ω we consider difference operators of te form 3) 5). We will need te following assumption. Assumption H 0 [f]. Suppose tat f CΩ, R) te derivatives x f = x f,..., xn f), p f, q f = q f,..., qn f) exist on Ω x f, q f CΩ, R n ), p f CΩ, R). Denote by z, u), u = u,..., u n ) te unknown functions of te variables t i), x m) ). Write u i,m) = u i,m),..., u i,m) n ) ) P i,m) [z, u] = t i), x m), z i,m), u i,m). We consider te system of difference equations ) 5) δ 0 z i,m) = f P i,m) [z, u] ) + qj f P i,m) [z, u] ) δ j z i,m) u i,m) j, 6) δ 0 u i,m) r = xr f P i,m) [z, u] ) + p f P i,m) [z, u] ) u i,m) r + qj f P i,m) [z, u] ) δ j u i,m) r, r =,..., n, wit te initial condition 7) z 0,m) = ϕ m), u0,m) = ψ m), N m N, were ϕ : E 0. R ψ = ψ.,..., ψ.n ): E 0. R n are given functions. Te operators δ 0 δ = δ,..., δ n ) are defined now in te following way. If te functions z u = u,..., u n ) are calculated on te set E [0, t i) ] R n ) ten we put 8) δ 0 z i,m) = 0 z i+,m) z i,m) ).

6 20 Te difference operators wit respect to te spatial variables are given in te following way: 9) δ j z i,m) = j z i,m+e j) z i,m)) if qj f P i,m) [z, u] ) 0, 20) δ j z i,m) = j z i,m) z i,m e j) ) if qj f P i,m) [z, u] ) < 0, were j =,..., n. Te difference expressions δ 0 u i,m) r, δ u i,m) r,..., δ n u i,m) r ), r n, are defined in te same way. Note tat if 0 M ten tere exists exactly one solution z, u ), z : E R, u = u.,..., u.n ) : E R n, of problem 5) 20). It is essential in our considerations tat we approximate solutions of nonlinear problem ), 2) wit solutions of te quasilinear difference system. More precisely: we will use 5) 7) for approximation of te solution v : E R of problem ), 2) te derivative x v : E R n. System 5), 6) is obtained in te following way. We first introduce an additional unknown function u = x z, u = u,..., u n ) in ). Ten we consider te following linearization of ) wit respect to u: 2) t zt, x) = f U ) + qj f U ) xj zt, x) u j t, x) ), were U = t, x, zt, x), ut, x)). By differentiating equation ) wit respect to x r, r n, we get te differential system in te unknown function u : 22) t u r t, x) = xr f U ) + p f U ) u r t, x) + qj f U ) xj u r t, x), r n. Assume tat x ϕ = x ϕ,..., xn ϕ ) exists on [ b, b]. It is natural to consider te following initial condition for system 2), 22): 23) z0, x) = ϕx), u0, x) = x ϕx), x [ b, b]. Difference problem 5) 7) is a discretization of system 2), 22) wit initial condition 23). In our approac, te discretization metod for system 2), 22) depends on te point of te mes on te previous values of z u.

7 2 2. Convergence of difference metods. We will denote by FX, Y ) te class of all functions defined on X taking values in Y, X Y being arbitrary sets. We will need te following assumption trougout te paper. Assumption H[f]. Suppose tat Assumption H 0 [f] is satisfied ) tere is A R + suc tat x f P ), p f P ), q f P ) A on Ω were P = t, x, p, q), 2) tere is B R + suc tat te terms x ft, x, p, q) x ft, x, p, q), p ft, x, p, q) p ft, x, p, q), q ft, x, p, q) q ft, x, p, q) are bounded from above by B [ p p + q q ]. Teorem 2.. Suppose tat Assumption H[f] is satisfied ), 0 M for P = t, x, p, q) Ω we ave 24) 0 j qj f P ) 0, 2) te function ϕ: [ b, b] R is of class C 2 v : E R is te solution of ), 2) v is of class C 2 on E, 3) z, u ) = z, u.,..., u.n ): E R +n is te solution of problem 5) 20) tere is α 0 : R suc tat ϕ m) ϕ m) + xϕ m) ψ m) α 0 ), N m N, lim 0 α 0 ) = 0. Ten tere is a function α: R + suc tat v i,m) z i,m) lim 0 α) = 0. + x v i,m) u i,m) α) on E Proof. Write w = x v w = w,..., w n ). Ten te functions v, w): E R +n are te solution of problem 2) 23). Let te functions be defined by 25) Γ.0 : E R, Γ : E Rn, Γ = Γ.,..., Γ.n ) Λ.0 : E R, Λ : E Rn, Λ = Λ.,..., Λ.n ) Γ i,m).0 = δ 0 v i,m) t v i,m) + qj f P i,m) [v, w] ) [ xj v i,m) δ j v i,m) ],

8 22 26) 27) Γ i,m).r Λ i,m) = δ 0 w r i,m) t w r i,m) + qj f P i,m) [v, w] ) [ xj w i,m) r ] δ j w r i,m), r =,..., n,.0 = f P i,m) [v, w] ) f P i,m) [z, u ] ) qj f P i,m) [v, w] ) w i,m) j + qj f P i,m) [z, u ] ) u i,m).j + [ ] qj f P i,m) [v, w] ) qj f P i,m) [z, u ] ) δ j v i,m), 28) Write Λ i,m).r = xr f P i,m) [v, w] ) + p f P i,m) [v, w] ) w i,m) r xr f P i,m) [z, u ] ) p f P i,m) [z, u ] )u i,m).r [ ] + qj f P i,m) [v, w] ) qj f P i,m) [z, u ] ) λ i,m) ξ i,m) = v i,m) z i,m) = w i,m) u i,m), λ i,m) =, λ i,m).,..., λi,m).n δ j w i,m) r, r =,..., n. It follows from 5), 6) from 2), 22) tat ξ λ satisfy te difference equations ξ i+,m) = ξ i,m) + 0 qj f P i,m) [z, u ] ) δ j ξ i,m) 29) 30) λ i+,m).r [ + 0 Γ i,m).0 + Λ i,m).0 = λ i,m).r [ Γ i,m).r + Λ i,m).r ], ). qj f P i,m) [z, u ] ) δ j λ i,m).r ], r =,..., n.

9 23 Let ω.0, ω : I R be te functions defined by 3) ω i).0 = max { ξi,m) : t i), x m) ) E }, 32) ω i) = max { λi,m) : t i), x m) ) E }, were 0 i N 0. We will write a difference inequality for te function ω.0 + ω. Put J + [i, m] = { j {,..., n} : qj f P i,m) [z, u ] ) 0 }, J [i, m] = {,..., n } \ J + [i, m]. Consider te operator W : FE, R) FE, R) defined by W [ξ] i,m) = ξ i,m) qj 0 f P i,m) [z, u ] ) j j J + [i,m] j J [i,m] j qj f P i,m) [z, u ] ) ξ i,m+e j) j qj f P i,m) [z, u ] ) ξ i,m e j) were ξ FE, R) t i), x m) ) E. It follows from 9), 20) 29), 30) tat [ 33) ξ i+,m) = W [ξ ] i,m) + 0 Λ i,m).0 + Γ i,m).0 ], t i), x m) ) E. For te function λ = λ.,..., λ.n ) we write ) W [λ ] i,m) = W [λ. ] i,m),..., W [λ.n ] i,m). According to 30) te definition of te difference operators δ,..., δ n ) we ave [ ] 34) λ i+,m) = W [λ ] i,m) + 0 Λ i,m) + Γ i,m), t i), x m) ) E. We conclude from Assumption H[f] from condition 2) of te teorem tat tere are functions γ 0, γ : R + a constant c R + suc tat 35) Γ i,m).0 γ 0), Γ i,m) γ), t i), x m) ) E, 36) xj vt, x) c, xj x r vt, x) c, t, x) E, j, r =,..., n, were lim γ 0) = 0, 0 lim γ) = 0. 0

10 24 According to Assumption H[f] 3), 32) we ave [ ] 37) Λ i,m).0 A + 2 cb) ω i).0 + ωi) + Aω i), [ ] 38) Λ i,m) B + 2 c) ω i).0 + ωi) + Aω i), were t i), x m) ) E. We conclude from 24) from 9), 20) tat 39) W [ξ ] i,m) ω i).0, ti), x m) ) E, W [λ ] i,m) 0 qj f P i,m) [z, u ] ) j 40) j J + [i,m] j J [i,m] λ i,m) qj f P i,m) [z, u ] ) λ i,m+e j) j qj f P i,m) [z, u ] ) λ i,m e j) ω i) j were t i), x m) ) E. It follows from 33) from Assumption H[f] tat 4) ω i+).0 ω i).0 [ + 0A + 2 cb)] A + cb) ω i) + 0γ 0 ), were 0 i N 0. In a similar way we obtain te difference inequality 42) ω i+) ω i) [ + 0B + 2 c) + 0 A] + 0 B + 2 c)ω i).0 + 0γ), were 0 i N 0. Write C = B + 3A + 4 cb. It follows from 4), 42) tat te difference inequality ω i+).0 + ω i+) is satisfied. Tis gives ) ω i).0 + ωi) + 0 C) + 0 [γ 0 ) + γ)], i = 0,,..., N 0, 43) ω i).0 + ωi) α), i = 0,,..., N 0, wit 44) α) = α 0 )e Ca + [γ 0 ) + γ)] eca C if C > 0, 45) α) = α 0 ) + [γ 0 ) + γ)] a if C = 0. Tis completes te proof of te teorem. Now we formulate a result on te error estimate for metod 5) 20).,

11 25 Lemma 2.2. Suppose tat all te assumptions of Teorem 2. are satisfied Ten ) te solution v : E R of ), 2) is of class C 3 on E, 2) te constant c R + is suc tat xj vt, x), xi x j vt, x), tt vt, x), ttxj vt, x), xi x j x r vt, x) c, were t, x) E i, j, r =,..., n. 46) v i,m) z i,m) on E were + x v i,m) u i,m) α) α) = α 0 )e ac + γ 0 ) eac if C > 0, C α) = α 0 ) + a γ 0 ) if C = 0, C = B + 3A + 4B c, γ 0 ) = 0 c [ + A M ]. Proof. It follows from assumption 2) tat estimates 35) old wit γ 0 ) = γ) = 2 γ 0). Ten we obtain te lemma from inequalities 43). Remark 2.3. If we apply metod 6), 7) to solve problem ), 2) numerically, ten we approximate derivatives wit respect to spatial variables wit difference expressions wic are calculated wit use of te previous values of te approximate solution. If we use metod 5) 7) ten we approximate te spatial derivatives of z wit using adequate difference equations wic are generated by te original problem. Terefore numerical results obtained by 5) 7) are better tan tose obtained by metod 6), 7). Remark 2.4. Results on te error estimates for metods 6), 7) 5) 7) can be caracterized as follows. In 2) 46) we ave estimated te terms v i,m) z i,m) v i,m) z i,m) + x v i,m) u i,m), respectively. Te functions ᾱ α in 2) 46) are similar. Terefore, numerical results obtained by 5) 7) for initial problem ), 2) are better tat tose obtained by 6), 7). We illustrate te above properties of difference metods by a numerical example.

12 26 Now we consider te system of difference equations 5), 6) wit operators δ 0 δ = δ,..., δ n ) defined by 3), 4) were t i), x m) ) E z : E R. Te difference expressions are defined in te same way. 47) δ 0 u i,m) r, δ u i,m) r,..., δ n u i,m) r ), r n, Teorem 2.5. Suppose tat Assumption H[f] is satisfied ), 0 M for P = t, x, p, q) Ω we ave n 0 j qj f P ) 0, j n, 2) te function ϕ: [ b, b] R is of class C 2 v : E R is te solution of ), 2) v is of class C 2 on E, 3) z, u ) = z, u.,..., u.n ): E R +n is te solution of problem 5) 7) wit δ 0 δ given by 3), 4), 4) tere is α 0 : R + suc tat ϕ m) ϕ m) + xϕ m) ψ m) α 0 ), N m N, lim 0 α 0 ) = 0. Ten tere is α: R + suc tat v i,m) z i,m) + x v i,m) u i,m) α) on E lim 0 α) = 0. Proof. Write w = x v, w = w,..., w n ). Ten te functions v, w): E R n satisfy 2) 23). Let te functions Γ.0, Γ = Γ.,..., Γ.n ), Λ.0, Λ = Λ.,..., Λ.n ), be defined by 25) 28) wit δ 0 δ = δ,..., δ n given by 3), 4). Write λ i,m) ξ i,m) = v i,m) z i,m) = w i,m) u i,m), λ i,m) =, λ i,m).,..., λi,m).n Suppose tat te functions ω.0, ω : I R + are defined by 3), 32) te operator W : FE, R) FE, R) is given in te following way: W [ξ] i,m) = [ 2 n + ] 0 qj f P i,m) [z, u ] ) ξ i,m+e j) j + [ 2 n ] 0 qj f P i,m) [z, u ] ) ξ i,m ej), j ).

13 27 were ξ FE, R) t i), x m) ) E. It follows tat relations 33), 34), 39), 40) are satisfied wit te above given W we get te difference inequality ω i+).0 + ω i+) ω i).0 + ωi) ) + 0C) + 0 [γ 0 ) + γ)], 0 i N 0, wit γ 0, γ, c satisfying 35), 36) C = B + 3A + 4 cb. Ten estimate 43) is satisfied wit α defined by 44), 45). Tis completes te proof. It it easy to formulate a result on te error estimate for te metod under te additional assumption tat te solution of ), 2) is of class C 3 on E. In te results on error estimates we need estimates for te derivatives of te solution v of problem ), 2). One may obtain tem by te metod of differential inequalities, see [5], Capter VII. 3. Numerical examples. Let n = E = { t, x) R 2 : t [0, ], x 2 2t }. Consider te differential equation 48) t zt, x) = 2 sin + xzt, x) ) + ft, x) wit te initial condition 49) z0, x) = 0, x [ 2, 2], were ft, x) = + x 3 2 sin + 3x2 t ). Te exact solution of tis problem is vt, x) = t+x 3 ), t, x) E. Te classical difference metod for 48), 49) as te form z i+,m) = [ ] z i,m+) + z i,m ) + 0 f i,m) 50) 2 + [ ] 0 2 sin + z i,m+) z i,m ) ) 2 ), 5) z 0,m) = 0 for x m) [ 2, 2], were f i,m) = ft i), x m) ). Note tat Teorem. does not apply to equation 48). Te convergence of metod 50), 5) follows from Teorem.3. Now we consider metod 5), 6) for problem 48), 49). Denote by z, u) te unknown functions of te variables t i), x m) ) consider te system of

14 28 difference equations 52) z i+,m) = z i,m) sin + ui,m) ) + 0 f i,m) + [ ] 0 2 cos + ui,m) )) δz i,m) u i,m), 53) u i+,m) = u i,m) + 0 F i,m) cos + ui,m) ) δu i,m) wit te initial condition 54) z 0,m) = 0, u 0,m) = 0, x m) [ 2, 2], were F i,m) = F t i), x m) ), F t, x) = 3x 2 3xt cos + 3x 2 t). Te difference expressions δz i,m) δu i,m) are defined in te following way. If cos + u i,m) ) 0 ten δz i,m) = zi,m+) z i,m) If cos + u i,m) ) < 0 ten δz i,m) = zi,m) z i,m ) δu i,m) = ui,m+) u i,m), δu i,m) = ui,m) u i,m ). Denote by z z, ũ ) te solutions of problems 50), 5) 52) 54), respectively. Consider te errors ε i,m) = v i,m) z i,m), ε i,m) = v i,m) z i,m), t i), x m) ) E. We put 0 = 0.00, = we ave te following experimental values for te errors ε ε. Table of errors, ε = v z t = 0.4 t = 0.5 t = 0.6 t = 0.7 x = x = x =

15 29 Table of errors, ε = v z t = 0.4 t = 0.5 t = 0.6 t = 0.7 x = x = x = Note tat εt, x) < εt, x) for all values of t, x). We also give te following information on te errors of metods 50), 5) 52) 54). Write η i) = max { ε i,m) : t i), x m) ) E }, η i) = max { ε i,m) : t i), x m) ) E }, 0 i N 0. In Table E, we give experimental values of te functions η η for 0 = 0.00, = Table E t = 0.40 t = 0.45 t = 0.50 t = 0.55 t = 0.60 t = 0.65 t = 0.70 ηt) : ηt) : Note tat ηt) < ηt) for all t. Tus we see tat te errors of metod 50), 5) are larger tan te errors of 52) 54). Tis is due to te fact tat te approximation of te spatial derivatives of z in 52) 54) is better tan te respective approximation of x z in 50), 5). Metods described in Teorems ave te potential for applications in te numerical solving of first order nonlinear differential equations.

16 30 References. Kowalski Z., A difference metod for te nonlinear partial differential equations, Ann. Polon. Mat., 8 966), , A difference metod for certain yperbolic systems of nonlinear partial differential equations of first order, Ann Polon. Mat., 9 967), , On te difference metod for certain yperbolic systems of nonlinear partial differential equations of te first order, Bull. Acad. Polon. Sci. Sér. Sci. Mat. Astronom. Pys., 6 968), Pliś A., On difference inequalities corresponding to partial differential inequalities of first order, Ann. Polon. Mat., ), Szarski J., Differential Inequalities, Polis Sci. Publ., Warszawa, 967. Received November 23, 200 Akademia Marynarki Wojennej Smidowicza Gdynia, Pol ABar@amw.gdynia.pl Uniwersytet Gdański Instytut Matematyki Wita Stwosza Gdańsk, Pol zkamont@mat.univ.gda.pl