3.7. REFERENCES 71
3.3. Show by a suitable transformation u D F .v/, the omas equation
u
xy
C au
x
u
y
C bu
x
C cu
y
D 0
(a; b and c constant) can be linearized.
3.4. Show that the forced Burgers’ equation
u
t
C 2uu
x
D u
xx
C f .x/
can be linearized via the Hopf–Cole transformation.
3.7 REFERENCES
[53] E. Hopf, e partial differential equation u
t
C uu
x
D u
xx
, Comm. Pure Appl. Math., 3,
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[54] J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Q. Appl. Math.,
9, pp. 225–236, 1951. DOI: 10.1090/qam/42889. 50
[55] A. R. Forsyth, eory of Differential Equations, vol. VI, p. 102, Ex. 3, Cambridge University
Press, 1906. 50
[56] D. J. Korteweg and F. de Vries, On the change of form of long waves advancing in a
rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39, pp. 422–
443, 1895. DOI: 10.1080/14786449508620739. 54
[57] R. M. Miura, e Korteweg–deVries Equation: A survey of results, SIAM Review, 18(3),
pp. 412–459, 1978. DOI: 10.1137/1018076. 54
[58] R. M. Miura, Korteweg–de Vries Equation and generalizations. I. A remarkable
explicit nonlinear transformation, J. Math. Phys., 9, pp. 1202–1204, 1968. DOI:
10.1063/1.1664700. 54
[59] R. M. Miura, C. S. Gardner, and M. D. Kruskal, Korteweg–de Vries Equation and gen-
eralizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 9,
pp. 1204–1209, 1968. DOI: 10.1063/1.1664701. 56
[60] G. M. Cox, J. M. Hill, and N. amwattana, A formal exact mathematical solution for a
sloping rat-hole in a highly frictional granular solid, Acta Mech., 170, pp. 127–147, 2004.
DOI: 10.1007/s00707-004-0118-x. 56
[61] D. J. Arrigo and F. Hickling, A Darboux transformation for a class of linear parabolic
partial differential equation, J. Phys. A: Math. Gen., 35, pp. 389–399, 2002. DOI:
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