Handbook of mathematics for engineers and scienteists part 186 ppt

T7.3. NONLINEAR EQUATIONS 1263 10. F 1  x, ∂w ∂x  = F 2  y, ∂w ∂y  . A separable equation. Complete integral: w = ϕ(x)+ψ(y)+C 1 , where the functions ϕ = ϕ(x)andψ = ψ(y) are determined from the ordinary differential equations F 1  x, ϕ  x  = C 2 , F 2  y, ψ  y  = C 2 . 11. F 1  x, ∂w ∂x  + F 2  y, ∂w ∂y  + aw =0. A separable equation. Complete integral: w = ϕ(x)+ψ(y), where the functions ϕ = ϕ(x)andψ = ψ(y) are determined from the ordinary differential equations F 1  x, ϕ  x  + aϕ = C 1 , F 2  y, ψ  y  + aψ =–C 1 , where C 1 is an arbitrary constant. If a ≠ 0, one can set C 1 = 0 in these equations. 12. F 1  x, 1 w ∂w ∂x  + w k F 2  y, 1 w ∂w ∂y  =0. A separable equation. Complete integral: w(x, y)=ϕ(x)ψ(y). The functions ϕ = ϕ(x)andψ = ψ(y) are determined by solving the ordinary differential equations ϕ –k F 1  x, ϕ  x /ϕ  = C, ψ k F 2  y, ψ  y /ψ  =–C, where C is an arbitrary constant. 13. F 1  x, ∂w ∂x  + e λw F 2  y, ∂w ∂y  =0. A separable equation. Complete integral: w(x, y)=ϕ(x)+ψ(y). The functions ϕ = ϕ(x)andψ = ψ(y) are determined by solving the ordinary differential equations e –λϕ F 1  x, ϕ  x  = C, e λψ F 2  y, ψ  y  =–C, where C is an arbitrary constant. 1264 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 14. F 1  x, 1 w ∂w ∂x  + F 2  y, 1 w ∂w ∂y  = k ln w. A separable equation. Complete integral: w(x, y)=ϕ(x)ψ(y). The functions ϕ = ϕ(x)andψ = ψ(y) are determined by solving the ordinary differential equations F 1  x, ϕ  x /ϕ  – k ln ϕ = C, F 2  y, ψ  y /ψ  – k ln ψ =–C, where C is an arbitrary constant. 15. ∂w ∂x + yF 1  x, ∂w ∂y  + F 2  x, ∂w ∂y  =0. Complete integral: w = ϕ(x)y –  F 2  x, ϕ(x)  dx + C 1 , where the function ϕ(x) is determined by solving the ordinary differential equation ϕ  x + F 1 (x, ϕ)=0. 16. F  ∂w ∂x + ay, ∂w ∂y + ax  =0. Complete integral: w =–axy + C 1 x + C 2 y + C 3 ,whereF (C 1 , C 2 )=0. 17.  ∂w ∂x  2 +  ∂w ∂y  2 = F  x 2 + y 2 , y ∂w ∂x – x ∂w ∂y  . Complete integral: w =–C 1 arctan y x + 1 2   ξF(ξ, C 1 )–C 2 1 dξ ξ +C 2 ,whereξ = x 2 +y 2 . 18. F  x, ∂w ∂x , ∂w ∂y  =0. Complete integral: w = C 1 y +ϕ(x, C 1 )+C 2 , where the function ϕ = ϕ(x, C 1 ) is determined from the ordinary differential equation F  x, ϕ  x , C 1  = 0. 19. F  ax + by, ∂w ∂x , ∂w ∂y  =0. For b = 0, see equation T7.3.3.18. Complete integral for b ≠ 0: w = C 1 x + ϕ(z, C 1 )+C 2 , z = ax + by, where the function ϕ = ϕ(z) is determined from the nonlinear ordinary differential equation F  z, aϕ  z + C 1 , bϕ  z  = 0. 20. F  w, ∂w ∂x , ∂w ∂y  =0. Complete integral: w = w(z), z = C 1 x + C 2 y, where C 1 and C 2 are arbitrary constants and w = w(z) is determined by the autonomous ordinary differential equation F  w, C 1 w  z , C 2 w  z  = 0. REFERENCES FOR CHAPTER T7 1265 21. F  ax + by + cw, ∂w ∂x , ∂w ∂y  =0. For c = 0, see equation T7.3.3.19. If c ≠ 0, then the substitution cu = ax + by + cw leads to an equation of the form T7.3.3.20: F  cu, ∂u ∂x – a c , ∂u ∂y – b c  = 0. 22. F  x, ∂w ∂x , ∂w ∂y , w – y ∂w ∂y  =0. Complete integral: w = C 1 y + ϕ(x), where the function ϕ(x) is determined from the ordinary differential equation F  x, ϕ  x , C 1 , ϕ  = 0. 23. F  w, ∂w ∂x , ∂w ∂y , x ∂w ∂x + y ∂w ∂y  =0. Complete integral: w = ϕ(ξ), ξ = C 1 x + C 2 y, where the function ϕ(ξ) is determined by solving the nonlinear ordinary differential equation F  ϕ, C 1 ϕ  ξ , C 2 ϕ  ξ , ξϕ  ξ  = 0. 24. F  ax + by, ∂w ∂x , ∂w ∂y , w – x ∂w ∂x – y ∂w ∂y  =0. Complete integral: w = C 1 x + C 2 y + ϕ(ξ), ξ = ax + by, where the function ϕ(ξ) is determined by solving the nonlinear ordinary differential equation F  ξ, aϕ  ξ + C 1 , bϕ  ξ + C 2 , ϕ – ξϕ  ξ  = 0. 25. F  x, ∂w ∂x , G  y, ∂w ∂y  =0. Complete integral: w = ϕ(x, C 1 )+ψ(y, C 1 )+C 2 , where the functions ϕ and ψ are determined by the ordinary differential equations F (x, ϕ  x , C 1 )=0, G(y, ψ  y )=C 1 . On solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate. References for Chapter T7 Kamke, E., Differentialgleichungen: L ¨ osungsmethoden und L ¨ osungen, II, Partielle Differentialgleichungen Erster Ordnung f ¨ ur eine gesuchte Funktion, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1965. Lopez, G., Partial Differential Equations of First Order and Their Applications to Physics, World Scientific Publishing Co., Singapore, 2000. Polyanin, A. D., Zaitsev, V. F., and Moussiaux, A., Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. Rhee, H., Aris, R., and Amundson, N. R., First Order Partial Differential Equations, Vols. 1 and 2, Prentice Hall, Englewood Cliffs, New Jersey, 1986 and 1989. Tran,D.V.,Tsuji,M.,andNguyen,D.T.S.,The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations, Chapman & Hall, London, 1999. Chapter T8 Linear Equations and Problems of Mathematical Physics T8.1. Parabolic Equations T8.1.1. Heat Equation ∂w ∂t = a ∂ 2 w ∂x 2 T8.1.1-1. Particular solutions. w(x)=Ax + B, w(x, t)=A(x 2 + 2at)+B, w(x, t)=A(x 3 + 6atx)+B, w(x, t)=A(x 4 + 12atx 2 + 12a 2 t 2 )+B, w(x, t)=x 2n + n  k=1 (2n)(2n – 1) (2n – 2k + 1) k! (at) k x 2n–2k , w(x, t)=x 2n+1 + n  k=1 (2n + 1)(2n) (2n – 2k + 2) k! (at) k x 2n–2k+1 , w(x, t)=A exp(aμ 2 t μx)+B, w(x, t)=A 1 √ t exp  – x 2 4at  + B, w(x, t)=A x t 3/2 exp  – x 2 4at  + B, w(x, t)=A exp(–aμ 2 t)cos(μx + B)+C, w(x, t)=A exp(–μx)cos(μx – 2aμ 2 t + B)+C, w(x, t)=A erf  x 2 √ at  + B, where A, B, C,andμ are arbitrary constants, n is a positive integer, and erf z ≡ 2 √ π  z 0 exp(–ξ 2 ) dξ is the error function (probability integral). T8.1.1-2. Formulas allowing the construction of particular solutions. Suppose w = w(x, t) is a solution of the heat equation. Then the functions w 1 = Aw( λx + C 1 , λ 2 t + C 2 )+B, w 2 = A exp(λx + aλ 2 t)w(x + 2aλt + C 1 , t + C 2 ), w 3 = A √ |δ + βt| exp  – βx 2 4a(δ + βt)  w  x δ + βt , γ + λt δ + βt  , λδ – βγ = 1, 1267 1268 LINEAR EQUATIONS AND PROBLEMS OF MATHEM ATI CAL PHYSICS where A, B, C 1 , C 2 , β, δ,andλ are arbitrary constants, are also solutions of this equation. The last formula with β = 1, γ =–1, δ = λ = 0 was obtained with the Appell transformation. T8.1.1-3. Cauchy problem and boundary value problems. For solutions of the Cauchy problem and various boundary value problems, see Subsec- tion T8.1.2 with Φ(x, t) ≡ 0. T8.1.2. Nonhomogeneous Heat Equation ∂w ∂t = a ∂ 2 w ∂x 2 + Φ(x, t) T8.1.2-1. Domain: –∞ < x < ∞. Cauchy problem. An initial condition is prescribed: w = f(x)att = 0. Solution: w(x, t)=  ∞ –∞ f(ξ)G(x, ξ,t) dξ +  t 0  ∞ –∞ Φ(ξ, τ )G(x, ξ, t – τ) dξ dτ, where G(x, ξ,t)= 1 2 √ πat exp  – (x – ξ) 2 4at  . T8.1.2-2. Solutions of boundary value problems in terms of the Green’s function. We consider boundary value problems on an interval 0 ≤ x ≤ l with the general initial condition w = f(x)att = 0 and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as w(x, t)=  l 0 f(ξ)G(x, ξ,t) dξ +  t 0  l 0 Φ(ξ, τ )G(x, ξ, t – τ) dξ dτ. Here, the upper limit l can be finite or infinite; if l = ∞, there is no boundary condition corresponding to it. Paragraphs T8.1.2-3 through T8.1.2-8 present the Green’s functions for various types of homogeneous boundary conditions. Remark. Formulas from Section 14.7 should be used to obtain solutions to corresponding nonhomoge- neous boundary value problems. T8.1.2-3. Domain: 0 ≤ x < ∞. First boundary value problem. A boundary condition is prescribed: w = 0 at x = 0. Green’s function: G(x, ξ,t)= 1 2 √ πat  exp  – (x – ξ) 2 4at  –exp  – (x + ξ) 2 4at  . T8.1. PARABOLIC EQUATIONS 1269 T8.1.2-4. Domain: 0 ≤ x < ∞. Second boundary value problem. A boundary condition is prescribed: ∂ x w = 0 at x = 0. Green’s function: G(x, ξ,t)= 1 2 √ πat  exp  – (x – ξ) 2 4at  +exp  – (x + ξ) 2 4at  . T8.1.2-5. Domain: 0 ≤ x < ∞. Third boundary value problem. A boundary condition is prescribed: ∂ x w – kw = 0 at x = 0. Green’s function: G(x, ξ, t)= 1 2 √ πat  exp  – (x – ξ) 2 4at  +exp  – (x + ξ) 2 4at  – 2k  ∞ 0 exp  – (x + ξ + η) 2 4at – kη  dη  . T8.1.2-6. Domain: 0 ≤ x ≤ l. First boundary value problem. Boundary conditions are prescribed: w = 0 at x = 0, w = 0 at x = l. Two forms of representation of the Green’s function: G(x, ξ,t)= 2 l ∞  n=1 sin  nπx l  sin  nπξ l  exp  – an 2 π 2 t l 2  = 1 2 √ πat ∞  n=–∞  exp  – (x – ξ + 2nl) 2 4at  –exp  – (x + ξ + 2nl) 2 4at  . The first series converges rapidly at large t and the second series at small t. T8.1.2-7. Domain: 0 ≤ x ≤ l. Second boundary value problem. Boundary conditions are prescribed: ∂ x w = 0 at x = 0, ∂ x w = 0 at x = l. Two forms of representation of the Green’s function: G(x, ξ,t)= 1 l + 2 l ∞  n=1 cos  nπx l  cos  nπξ l  exp  – an 2 π 2 t l 2  = 1 2 √ πat ∞  n=–∞  exp  – (x – ξ + 2nl) 2 4at  +exp  – (x + ξ + 2nl) 2 4at  . The first series converges rapidly at large t and the second series at small t. . G., Partial Differential Equations of First Order and Their Applications to Physics, World Scientific Publishing Co., Singapore, 2000. Polyanin, A. D., Zaitsev, V. F., and Moussiaux, A., Handbook. Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. Rhee, H., Aris, R., and Amundson, N. R., First Order Partial Differential Equations, Vols. 1 and 2, Prentice Hall,. Englewood Cliffs, New Jersey, 1986 and 1989. Tran,D.V.,Tsuji,M.,andNguyen,D.T.S.,The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations, Chapman