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Mesh-free methods and time integrations for transient heat conduction

Sladek J., Sladek V., Zhang C.

Computer Assisted Mechanics and Engineering Sciences

2011

Vol. 18, no. 1-2

15-128

The paper deals with transient heat conduction in functionally gradient materials. The spatial variation of the temperature field is approximated by using alternatively two various mesh free approximations,while the time dependence is treated either by the Laplace transform method and/or by the polynomialinterpolation in the time stepping method. The accuracy and convergence of the numerical results as well as the computational e?ciency of various approaches are compared in numerical test example.

Numerical solution of portfolio optimal control with constraints: the case of the object as weakly singular integral equation

Filatova D., Grzywaczewski M.

Journal of Applied Computer Science

2007

Vol. 15, nr 2

51-69

We consider the model of optimal portfolio of Mertons’ market model. The noises involved in the dynamics of the wealth are fractional white noises. The stochastic optimal control problem is converted into a non-random optimization. An example of problem numerical solution illustrates proposed methodology.

Non-linear m-parabolic problem for some time-spatial domain in En+1

Filar M., Szałajko K.

Opuscula Mathematica

1999

Vol. 19

63-73

The aim of this paper is to find the solution u of non-linear m-parabolic equation Pmu(X) = F(X, P0u(X),..., Pm-1u(X)), X = (x,t), x = (x1, ..., xn) ∈ En in the domain D = D0x (0,T], D0 = {x: xi > 0, i = 1, ..., n}, T > 0 is a constant, satisfying the conditions Piu(X) = 0 for X ∈ D0 x {0}, i = 0,1,...,m-1, Piu(X) = 0 for X ∈ Dj x (0,T], i = 0,1, ..., m - 1; j = 1,2, ..., n, Dk(xj) Piu(X) are bounded on D for i = 0,1, ..., m - 1; j = 1,2, ...,n; k = 0, 1, Dj = {x: xj = 0, xk > 0, k ∈ {1,2, ..., n} \ {j}}, j = 1, ..., n. The function u is the solution of suitable system of integral equations. The existence and uniqueness of the solution of this system follows from the fix point Banach theorem.

Celem pracy jest znalezienie rozwiązania u równania m-parabolicznego Pmu(X) = F(X, P0u(X),..., Pm-1u(X)), X = (x,t), x = (x1, ..., xn) ∈ En w obszarze D = D0 x (0,T], D0 = {x: xi > 0,i = 1, ..., n}, T > 0 jest stałą, spełniającego warunki Piu(X) = 0 dla X ∈ D0 x {0}, i = 0,1,..., m-1, Piu(X) = 0 dla X ∈ Dj x (0,T], i = 0,1, ..., m - 1; j = 1,2, ..., n, Dk(xj)Piu(X) są ograniczone w D dla i = 0,1, ..., m - 1; j = 1,2, ..., n; k = 0,1, Dj = {x: xj = 0, xk > 0, k ∈ {1,2, ..., n} \ {j}}, j = 1, ..., n. Szukana funkcja u jest rozwiązaniem stosownego układu równań całkowych. Istnienie i jednoznaczność rozwiązania tego układu wynika z twierdzenia Banacha o punkcie stałym.