1298 LINEAR EQUATIONS AND PROBLEMS OF MATHEM ATICAL PHYSICS T8.4.3-4. Boundary value problem for a circle. Domain: 0 ≤ r ≤ a. Boundary conditions in the polar coordinate system: w = f(ϕ)atr = a, ∂ r w = g(ϕ)atr = a. Solution: w(r, ϕ)= 1 2πa (r 2 –a 2 ) 2 2π 0 [a –r cos(η –ϕ)]f(η) dη [r 2 +a 2 –2ar cos(η – ϕ)] 2 – 1 2 2π 0 g(η) dη r 2 +a 2 –2ar cos(η –ϕ) . T8.4.4. Nonhomogeneous Biharmonic Equation ΔΔw = Φ(x, y) T8.4.4-1. Domain: –∞ < x < ∞,–∞ < y < ∞. Solution: w(x, y)= ∞ –∞ ∞ –∞ Φ(ξ, η) (x – ξ, y – η) dξ dη, (x, y)= 1 8π (x 2 + y 2 )ln x 2 + y 2 . T8.4.4-2. Domain: –∞ < x < ∞, 0 ≤ y < ∞. Boundary value problem. The upper half-plane is considered. The derivatives are prescribed at the boundary: ∂ x w = f(x)aty = 0, ∂ y w = g(x)aty = 0. Solution: w(x, y)= 1 π ∞ –∞ f(ξ) arctan x – ξ y + y(x – ξ) (x – ξ) 2 + y 2 dξ + y 2 π ∞ –∞ g(ξ) dξ (x – ξ) 2 + y 2 + 1 8π ∞ –∞ dξ ∞ 0 1 2 (R 2 + – R 2 – )–R 2 – ln R + R – Φ(ξ, η) dη + C, where C is an arbitrary constant, R 2 + =(x – ξ) 2 +(y + η) 2 , R 2 – =(x – ξ) 2 +(y – η) 2 . T8.4.4-3. Domain: 0 ≤ x ≤ l 1 , 0 ≤ y ≤ l 2 . The sides of the plate are hinged. A rectangle is considered. Boundary conditions are prescribed: w = ∂ xx w = 0 at x = 0, w = ∂ xx w = 0 at x = l 1 , w = ∂ yy w = 0 at y = 0, w = ∂ yy w = 0 at y = l 2 . Solution: w(x, y)= l 1 0 l 2 0 Φ(ξ, η)G(x, y, ξ, η) dη dξ, where G(x, y, ξ, η)= 4 l 1 l 2 ∞ n=1 ∞ m=1 1 (p 2 n + q 2 m ) 2 sin(p n x)sin(q m y)sin(p n ξ)sin(q m η), p n = πn l 1 , q m = πm l 2 . REFERENCES FOR CHAPTER T8 1299 References for Chapter T8 Butkovskiy,A.G.,Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984. Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, London, 1977. Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. Sutton, W. G. L., On the equation of diffusion in a turbulent medium, Proc. Roy. Soc., Ser. A, Vol. 138, No. 988, pp. 48–75, 1943. Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publications, New York, 1990. Chapter T9 Nonlinear Mathematical Physics Equations T9.1. Parabolic Equations T9.1.1. Nonlinear Heat Equations of the Form ∂w ∂t = ∂ 2 w ∂x 2 + f(w) Equations of this form admit traveling-wave solutions w = w(z), z = kx + λt,where k and λ are arbitrary constants, and the function w(z) is determined by the second-order autonomous ordinary differential equation ak 2 w zz – λw z + f(w)=0. 1. ∂w ∂t = ∂ 2 w ∂x 2 + aw(1–w). Fisher’s equation. This equation arises in heat and mass transfer, biology, and ecology. Traveling-wave solutions (C is an arbitrary constant): w(x, t)= 1 + C exp – 5 6 at 1 6 √ 6ax –2 , w(x, t)= 1 + 2C exp – 5 6 at 1 6 √ –6ax 1 + C exp – 5 6 at 1 6 √ –6ax 2 . 2. ∂w ∂t = ∂ 2 w ∂x 2 + aw – bw 3 . 1 ◦ . Solutions with a > 0 and b > 0: w(x, t)= a b C 1 exp 1 2 √ 2ax – C 2 exp – 1 2 √ 2ax C 1 exp 1 2 √ 2ax + C 2 exp – 1 2 √ 2ax + C 3 exp – 3 2 at , w(x, t)= a b 2C 1 exp √ 2ax + C 2 exp 1 2 √ 2ax– 3 2 at C 1 exp √ 2ax + C 2 exp 1 2 √ 2ax– 3 2 at + C 3 – 1 , where C 1 , C 2 ,andC 3 are arbitrary constants. 2 ◦ . Solutions with a < 0 and b > 0: w(x, t)= |a| b sin 1 2 √ 2|a| x + C 1 cos 1 2 √ 2|a| x + C 1 + C 2 exp – 3 2 at . 3 ◦ . Solutions with a = 0 and b > 0: w(x, t)= 2 b 2C 1 x + C 2 C 1 x 2 + C 2 x + 6C 1 t + C 3 . 1301 1302 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS 4 ◦ . Solution with a = 0 (generalizes the solution of Item 3 ◦ ): w(x, y)=xu(z), z = t + 1 6 x 2 , where the function u(z) is determined by the autonomous ordinary differential equation u zz – 9bu 3 = 0. 5 ◦ .Fora = 0, there is a self-similar solution of the form w(x, t)=t –1/2 f(ξ), ξ = xt –1/2 , where the function f(ξ) is determined by the ordinary differential equation f ξξ + 1 2 ξf ξ + 1 2 f – bf 3 = 0. 3. ∂w ∂t = ∂ 2 w ∂x 2 – w(1–w)(a – w). Fitzhugh–Nagumo equation. This equation arises in genetics, biology, and heat and mass transfer. Solutions: w(x, t)= A exp(z 1 )+aB exp(z 2 ) A exp(z 1 )+B exp(z 2 )+C , z 1 = √ 2 2 x + 1 2 – a t, z 2 = √ 2 2 ax + a 1 2 a – 1 t, where A, B,andC are arbitrary constants. 4. ∂w ∂t = ∂ 2 w ∂x 2 + aw + bw m . 1 ◦ . Traveling-wave solutions (the signs are chosen arbitrarily): w(x, t)= β + C exp(λt μx) 2 1–m , where C is an arbitrary constant and β = – b a , λ = a(1 – m)(m + 3) 2(m + 1) , μ = a(1 – m) 2 2(m + 1) . 2 ◦ .Fora = 0, there is a self-similar solution of the form w(x, t)=t 1/(1–m) U(z), where z = xt –1/2 . 5. ∂w ∂t = ∂ 2 w ∂x 2 + a + be λw . Traveling-wave solutions (the signs are chosen arbitrarily): w(x, t)=– 2 λ ln β + C exp μx – 1 2 aλt , β = – b a , μ = aλ 2 , where C is an arbitrary constant. T9.1. PARABOLIC EQUATIONS 1303 6. ∂w ∂t = ∂ 2 w ∂x 2 + aw ln w. Functional separable solutions: w(x, t)=exp Ae at x + A 2 a e 2at + Be at , w(x, t)=exp 1 2 – 1 4 a(x + A) 2 + Be at , w(x, t)=exp – a(x + A) 2 4(1 + Be –at ) + 1 2B e at ln(1 + Be –at )+Ce at , where A, B,andC are arbitrary constants. T9.1.2. Equations of the Form ∂w ∂t = ∂ ∂x f(w) ∂w ∂x + g(w) Equations of this form admit traveling-wave solutions w = w(z), z = kx + λt,where k and λ are arbitrary constants and the function w(z) is determined by the second-order autonomous ordinary differential equation k 2 [f(w)w z ] z – λw z + f(w)=0. 1. ∂w ∂t = a ∂ ∂x w m ∂w ∂x . This equation occurs in nonlinear problems of heat and mass transfer and flows in porous media. 1 ◦ . Solutions: w(x, t)=( kx + kλt + A) 1/m , k =λm/a, w(x, t)= m(x – A) 2 2a(m + 2)(B – t) 1 m , w(x, t)= A|t + B| – m m+2 – m 2a(m + 2) (x + C) 2 t + B 1 m , w(x, t)= m(x + A) 2 ϕ(t) + B|x + A| m m+1 |ϕ(t)| – m(2m+3) 2(m+1) 2 1 m , ϕ(t)=C – 2a(m + 2)t, where A, B, C,andλ are arbitrary constants. The second solution for B > 0 corresponds to blow-up regime (the solution increases without bound on a finite time interval). 2 ◦ . There are solutions of the following forms: w(x, t)=(t + C) –1/m F (x) (multiplicative separable solution); w(x, t)=t λ G(ξ), ξ = xt – mλ+1 2 (self-similar solution); w(x, t)=e –2λt H(η), η = xe λmt (generalized self-similar solution); w(x, t)=(t + C) –1/m U(ζ), ζ = x + λ ln(t + C), where C and λ are arbitrary constants. 1304 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS 2. ∂w ∂t = a ∂ ∂x w m ∂w ∂x + bw. By the transformation w(x, t)=e bt v(x, τ), τ = 1 bm e bmt + C the original equation can be reduced to an equation of the form T9.1.2.1: ∂v ∂τ = a ∂ ∂x v m ∂v ∂x . 3. ∂w ∂t = a ∂ ∂x w m ∂w ∂x + bw m+1 . 1 ◦ . Multiplicative separable solution (a = b = 1, m > 0): w(x, t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2(m + 1) m(m + 2) cos 2 (πx/L) (t 0 – t) 1/m for |x| ≤ L 2 , 0 for |x| > L 2 , where L = 2π(m + 1) 1/2 /m. This solution describes a blow-up regime that exists on a limited time interval t [0, t 0 ). The solution is localized in the interval |x| < L/2. 2 ◦ . Multiplicative separable solution: w(x, t)= Ae μx + Be –μx + D mλt + C 1/m , B = λ 2 (m + 1) 2 4b 2 A(m + 2) 2 , D =– λ(m + 1) b(m + 2) , μ = m – b a(m + 1) , where A, C,andλ are arbitrary constants, ab(m + 1)<0. 3 ◦ . Functional separable solutions [it is assumed that ab(m + 1)<0]: w(x, t)= F (t)+C 2 |F (t)| m+2 m+1 e λx 1/m , F (t)= 1 C 1 – bmt , λ = m –b a(m + 1) , where C 1 and C 2 are arbitrary constants. 4 ◦ . There are functional separable solutions of the following forms: w(x, t)= f(t)+g(t)(Ae λx + Be –λx ) 1/m , λ = m –b a(m + 1) ; w(x, t)= f(t)+g(t)cos(λx + C) 1/m , λ = m b a(m + 1) , where A, B,andC are arbitrary constants. . Conduction of Heat in Solids, Clarendon Press, Oxford, 1984. Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, London, 1977. Polyanin, A. D., Handbook of Linear Partial Differential. q m = πm l 2 . REFERENCES FOR CHAPTER T8 1299 References for Chapter T8 Butkovskiy,A.G.,Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982. Carslaw, H. S. and Jaeger,. Nonlinear Heat Equations of the Form ∂w ∂t = ∂ 2 w ∂x 2 + f(w) Equations of this form admit traveling-wave solutions w = w(z), z = kx + λt,where k and λ are arbitrary constants, and the function w(z)