Burgers’ Equation Is . . . Enter Symmetries Let Us Use Symmetries Why Burgers Equation: What Are the . . . Can Burgers’ Equation . . . Symmetry-Based Approach Analysis of the Problem Conclusion: Which . . . Leobardo Valera, Martine Ceberio, Excluding Additional . . . and Vladik Kreinovich Home Page Computational Science Program Title Page University of Texas at El Paso ◭◭ ◮◮ 500 W. University El Paso, TX 79968, USA ◭ ◮ leobardovalera@gmail.com, mceberio@utep.edu, vladik@utep.edu Page 1 of 18 Go Back Full Screen Close Quit
Burgers’ Equation Is . . . 1. Burgers’ Equation Is Ubiquitous Enter Symmetries Let Us Use Symmetries • In many application areas we encounter the Burgers’ What Are the . . . equation Can Burgers’ Equation . . . ∂x = d · ∂ 2 u ∂u ∂t + u · ∂u ∂x 2 . Analysis of the Problem Conclusion: Which . . . • Our interest in this equation comes from the use of Excluding Additional . . . these equations for describing shock waves. Home Page • Its applications range from fluid dynamics to nonlinear Title Page acoustics, gas dynamics, and dynamics of traffic flows. ◭◭ ◮◮ • This seems to indicate that this equation ◭ ◮ – reflects some fundamental ideas, Page 2 of 18 – and not just ideas related to liquid or gas dynamics. Go Back • In this talk, we show that indeed, the Burgers’ equation Full Screen can be deduced from fundamental principles. Close Quit
Burgers’ Equation Is . . . 2. Enter Symmetries Enter Symmetries Let Us Use Symmetries • How do we make predictions in general? What Are the . . . • We observe that, in several situations, a body left in Can Burgers’ Equation . . . the air fell down. Analysis of the Problem Conclusion: Which . . . • We thus conclude that in similar situations, a body will Excluding Additional . . . also fall down. Home Page • Behind this conclusion is the fact that there is some Title Page similarity between the new and the old situations. ◭◭ ◮◮ • In other words, there are transformations that: ◭ ◮ – transform the old situation into a new one Page 3 of 18 – under which the physics will be mostly preserved, Go Back – i.e., which form what physicists call symmetries . Full Screen Close Quit
Burgers’ Equation Is . . . 3. Symmetries (cont-d) Enter Symmetries Let Us Use Symmetries • In the falling down example: What Are the . . . – we can move to a new location, we can rotate around, Can Burgers’ Equation . . . – the falling process will remain. Analysis of the Problem Conclusion: Which . . . • Thus, shifts and rotations are symmetries of the falling- Excluding Additional . . . down phenomena. Home Page • In more complex situations, the behavior of a system Title Page may change with shift or with rotation. ◭◭ ◮◮ • However, the equations describing such behavior re- ◭ ◮ main the same. Page 4 of 18 Go Back Full Screen Close Quit
Burgers’ Equation Is . . . 4. Let Us Use Symmetries Enter Symmetries Let Us Use Symmetries • Symmetries are the fundamental reason why we are What Are the . . . capable of predictions. Can Burgers’ Equation . . . • Not surprisingly, symmetries have become one of the Analysis of the Problem main tools of modern physics. Conclusion: Which . . . • Let us therefore use symmetries to explain the ubiquity Excluding Additional . . . of the Burgers’ equation. Home Page Title Page • Which symmetries should we use? ◭◭ ◮◮ • Numerical values of physical quantities depend on the measuring unit. ◭ ◮ • For example, when we measure distance x first in me- Page 5 of 18 ters and then in centimeters: Go Back – the quantity remains the same, Full Screen – but it numerical values change: instead of the orig- inal value x , we get x ′ = λ · x for λ = 100. Close Quit
Burgers’ Equation Is . . . 5. Let Us Use Symmetries (cont-d) Enter Symmetries Let Us Use Symmetries • In many cases, there is no physically selected unit of What Are the . . . length. Can Burgers’ Equation . . . • In such cases, it is reasonable to require that the cor- Analysis of the Problem responding physical equations be invariant Conclusion: Which . . . Excluding Additional . . . – with respect to such change of measuring unit, Home Page – i.e., with respect to the transformation x → x ′ = Title Page λ · x . ◭◭ ◮◮ • Of course, once we change the unit for measuring x , we may need to change related units; for example: ◭ ◮ – if we change a unit of current I in Ohm’s formula Page 6 of 18 V = I · R , Go Back – for the equation to remain valid we need to also Full Screen appropriately change, e.g., the unit for voltage V . Close Quit
Burgers’ Equation Is . . . 6. Let Us Use Symmetries (cont-d) Enter Symmetries Let Us Use Symmetries • In our case, there seems to be no preferred measuring What Are the . . . unit. Can Burgers’ Equation . . . • So, it is reasonable to require that: Analysis of the Problem Conclusion: Which . . . – the corresponding equation be invariant under trans- Excluding Additional . . . formations x → λ · x Home Page – if we appropriately change measuring units for all Title Page other quantities. ◭◭ ◮◮ ◭ ◮ Page 7 of 18 Go Back Full Screen Close Quit
Burgers’ Equation Is . . . 7. What Are the Symmetries of the Burgers’ Equa- Enter Symmetries tion Let Us Use Symmetries What Are the . . . • We want to check if, for every λ , Can Burgers’ Equation . . . – once we combine the re-scaling x → x ′ = λ · x with Analysis of the Problem the appropriate re-scalings Conclusion: Which . . . t → t ′ = a ( λ ) · t and u → u ′ = b ( λ ) · u, Excluding Additional . . . Home Page – the Burgers’ equation will preserve its form. Title Page • Let us keep only the time derivative in the left-hand ◭◭ ◮◮ side of the equation: ◭ ◮ ∂x + d · ∂ 2 u ∂u ∂t = − u · ∂u ∂x 2 . Page 8 of 18 Go Back • Then, the time derivative is described as a function on Full Screen the current values of u . Close Quit
Burgers’ Equation Is . . . 8. Symmetries of the Burgers’ Equation (cont-d) Enter Symmetries Let Us Use Symmetries • After the transformation, e.g., the partial derivative What Are the . . . ∂t is multiplied by b ( λ ) ∂u a ( λ ): Can Burgers’ Equation . . . Analysis of the Problem ∂u ′ ∂t ′ = b ( λ ) a ( λ ) · ∂u Conclusion: Which . . . ∂t . Excluding Additional . . . Home Page • More generally, the equation gets transformed into the following form: Title Page λ 2 · ∂ 2 u a ( λ ) · ∂u b ( λ ) ∂t = − b ( λ ) · b ( λ ) · u · ∂u ∂x + d · b ( λ ) ◭◭ ◮◮ ∂x 2 . λ ◭ ◮ • Dividing both sides of this equation by the coefficient Page 9 of 18 b ( λ ) a ( λ ) at the time derivative, we conclude that Go Back Full Screen λ 2 · ∂ 2 u ∂t = − b ( λ ) · a ( λ ) ∂x + d · a ( λ ) ∂u · u · ∂u ∂x 2 . Close λ Quit
Burgers’ Equation Is . . . 9. Symmetries of the Burgers’ Equation (cont-d) Enter Symmetries Let Us Use Symmetries • By comparing the two equations, we conclude that they What Are the . . . are equivalent Can Burgers’ Equation . . . – if the coefficients at the two terms in the right-hand Analysis of the Problem side are the same, Conclusion: Which . . . – i.e., if b ( λ ) · a ( λ ) = 1 and a ( λ ) Excluding Additional . . . = 1. λ 2 λ Home Page • The second equality implies that a ( λ ) = λ 2 , and the Title Page λ a ( λ ) = λ − 1 . first one, that b ( λ ) = ◭◭ ◮◮ ◭ ◮ • Thus, the Burgers’ equation is invariant under the trans- formation x → λ · x , t → λ 2 · t , and u → λ − 1 · u . Page 10 of 18 Go Back Full Screen Close Quit
Burgers’ Equation Is . . . 10. Can Burgers’ Equation Be Uniquely Deter- Enter Symmetries mined by Its Symmetries? Let Us Use Symmetries What Are the . . . • Let us consider a general equation in which the time Can Burgers’ Equation . . . derivative of u depends on the current values of u : Analysis of the Problem ∂x, ∂ 2 u � � ∂u u, ∂u Conclusion: Which . . . ∂t = f ∂x 2 , . . . . Excluding Additional . . . Home Page • Here, we assume that the function f is analytical, i.e., that it can be expanded into Taylor series Title Page ◭◭ ◮◮ ∂x, ∂ 2 u � � u, ∂u f ∂x 2 , . . . = ◭ ◮ � i 2 � i k � i 1 Page 11 of 18 � ∂ 2 u � ∂ k u � ∂u a i 1 ...i k · u i 0 · � · · . . . · , ∂x 2 ∂x k ∂x Go Back i 0 ,i 1 ,...,i k Full Screen where i 0 , i 1 , . . . , i k ∈ N . Close Quit
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