Numerische Simulation von Diffusionsprozessen mit nichtnegativitätserhaltenden konservativen Differenzenverfahren /

1982 ◽  
Vol 37 (8) ◽  
Author(s):  
Rudolf Gorenflo ◽  
Angelika Kuban
Keyword(s):  
Heat Equation ◽  
Linear Equation ◽  
Source Term ◽  
Difference Schemes ◽  
General Linear ◽  
Two Dimensional ◽  
Numerische Simulation ◽  
General Linear Equation ◽  
Conservative Difference Schemes ◽  
Conservative Difference

After analyzing the general linear equation of diffusion (of a substance or of energy) with source term and given influx across the boundary we describe a method for constructing explicit and implicit conservative difference schemes which also preserve nonnegativity. We call a scheme “conservative” if via a convenient sum-analogue it does exactly imitate the conservation of a substance or energy. We concretize this method for the spatially two-dimensional heat equation in a rectangle with given influx. We also present a conservative implicit scheme with alternating directions

Numerische Simulation von Diffusionsprozessen mit nichtnegativitätserhaltenden konservativen Differenzenverfahren / Numerical Simulation of Diffusion Processes with conservative difference schemes of non-negative type

1982 ◽  
Vol 37 (8) ◽  
pp. 759-768 ◽  
Author(s):  
Rudolf Gorenflo ◽  
Angelika Kuban
Keyword(s):  
Numerical Simulation ◽  
Heat Equation ◽  
Source Term ◽  
Diffusion Processes ◽  
Difference Schemes ◽  
Two Dimensional ◽  
Numerische Simulation ◽  
General Linear Equation ◽  
Conservative Difference Schemes ◽  
Conservative Difference

After analyzing the general linear equation of diffusion (of a substance or of energy) with source term and given influx across the boundary we describe a method for constructing explicit and implicit conservative difference schemes which also preserve nonnegativity. We call a scheme “conservative” if via a convenient sum-analogue it does exactly imitate the conservation of a substance or energy. We concretize this method for the spatially two-dimensional heat equation in a rectangle with given influx. We also present a conservative implicit scheme with alternating directions

scholarly journals On a general bilinear functional equation

2021 ◽  
Author(s):  
Anna Bahyrycz ◽  
Justyna Sikorska
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
General Linear ◽  
Linear Spaces ◽  
General Linear Equation ◽  
Bilinear Functional

AbstractLet X, Y be linear spaces over a field $${\mathbb {K}}$$ K . Assume that $$f :X^2\rightarrow Y$$ f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all $$x,x_i,y,y_i \in X$$ x , x i , y , y i ∈ X and with $$a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}$$ a i , b i ∈ K \ { 0 } , $$A_i,\,B_i \in {\mathbb {K}}$$ A i , B i ∈ K ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all $$x_i,y_i \in X$$ x i , y i ∈ X ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ), where $$C_1:=A_1B_1$$ C 1 : = A 1 B 1 , $$C_2:=A_1B_2$$ C 2 : = A 1 B 2 , $$C_3:=A_2B_1$$ C 3 : = A 2 B 1 , $$C_4:=A_2B_2$$ C 4 : = A 2 B 2 . We describe the form of solutions and study relations between $$(*)$$ ( ∗ ) and $$(**)$$ ( ∗ ∗ ) .

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