Weak Solution of the Singular Cauchy Problem of Euler-Poisson-Darboux Equation for n =4

Transcription

1 Applied Mathematical Sciences, Vol. 7, 3, no. 7, Weak Solution of the Singular Cauchy Problem of Euler-Poisson-Darboux Equation f n =4 A. Manyonge, D. Kweyu, J. Bitok 3, H. Nyambane 3 and J. Maremwa 3 Department of Pure and Applied Mathematics Maseno University, Maseno, Kenya wmanyonge@gmail.com Department of Mathematics Kenyatta University, Nairobi, Kenya kweyudavid@yahoo.com 3 Department of Mathematics and Computer Science Chepkoilel University College, Eldet, Kenya nyamabane.haron@yahoo.com, jmaremwa@gmail.com Abstract We consider the singular Cauchy problem of Euler-Poisson-Darboux equationepd of the fm u tt + k t u t = u ux, = fx,u t x, = where is the Laplacian operat in R n, n will refer to dimension and k a real parameter. The EPD equation finds applications in geometry, applied mathematics, physics etc. Various auths have solved this problem f different values of n and k using various techniques since the time of Euler[]. In this paper, we shall take the Fourier transfm of the EPD equation with respect to the space codinate. The equation so obtained is transfmed into a Bessel differential equation. We solve this equation and obtain the inverse Fourier transfm of its solution. Finally on using the convolution theem, we obtain the weak solution of the EPD equation f n = 4. The case f n = is an interesting one f this problem as the solution will be that of a D wave equation. We therefe deduce the weak solution f the D wave equation as well.

2 36 A. Manyonge, D. Kweyu, J. Bitok, H. Nyambane and J. Maremwa Mathematics Subject Classification: 35Q5 Keywds: Singular Cauchy problem, EPD, Bessel function, weak solution Introduction The study of singular Cauchy problems of the Euler-Poisson-Darboux EPD equation is concerned with the equation u tt + k t u t = u ux, = fx,u t x, = where is the Laplacian operat in R n, n will refer to dimension, x = x,x,...,x n is a point in R n, k is a real parameter and t is the time variable. Problems of type will be called singular if the coefficient k as t t and degenerate if k ast in such a way as not to change the t type of problem. In this paper, we shall study singular Cauchy problem of the EPD type. There has been several approaches by different people to the question of obtaining the solution of the singular Cauchy problems f the EPD equation. Our main objective is to obtain a weak solution of the EPD equation f a general n and deduce solution f n = 4. The case f n = is an interesting one f this problem as the solution will be that of a D wave equation. We therefe further deduce the weak solution of the D wave equation. The EPD equation f special values of k and n has occurred in many classical problems in geometry, applied mathematics and physics f over two centuries. Euler[] first considered the EPD equation f n =. Using the Riemann method, Martin[] gave the solution of f n = and k =,, 3... Diaz and Weinberger[3] obtained solutions of f k = n, n +,n+... from the known solution f k = n. They directly verified that the resulting fmula gives a solution of the problem f any k with Rek >n. Blum[4] obtained solution of the EPD f exceptional cases k =, 3, Weinstein[5] gave a complete solution of singular Cauchy problem covering all values of k and n. Blum s solution differes from the solution of Weinstein in that the function f is required to have fewer continuous derivatives i.e. it is sufficient f f to have derivatives of der of at least n k+3. Fusaro[6] solved the boundary value problem of the singular Cauchy problem of the EPD by separation of variables. Dernek[7], by using a

3 Weak solution of singular Cauchy problem 37 series, obtained the solution of the EPD of the fm u tt +at + b t u t = u, ux, = fx, u t x, = where a and b are real constants and fx is an initial function.uchikoshi[8] studied local Cauchy problems in a complex domain f the EPD of an incompressible fluid. In this paper, our wk considers the weak solution of the singular Cauchy problem of the Euler-poisson-Darboux equation f n 4 and deduce a weak solution of the D wave equation. Mathematical Methodology. Weak solution f n=4 In let k = n. This choice of k ensures that we are dealing only with singular Cauchy problems. F this, the EPD assumes the fm u t + n u t t = u 3 x Definition. The Fourier transfm of a function fx onr n denoted F is defined by F ξ =F[fx] = fxe iξx dx n Taking the Fourier transfm with respect to space codinate x f x R of both sides of 3 and assuming that ux, t and u both asx t we find Let d F[u] + n dt t df[u] dt + ξ F[u] = 4 n F[u] =F[u]ξ,t=t n V θ 5 where θ = ξ t. Now df[u] dt { n = t n ξ dv dθ n } V t and { d F[u] = t n dt n ξ d V dθ n ξ dv t dθ +n n } V t

4 38 A. Manyonge, D. Kweyu, J. Bitok, H. Nyambane and J. Maremwa On substitution of the above expressions into 4 we find ξ d V dθ + ξ dv ξ t dθ + d V dθ + dv ξ t dθ + d V dθ d V dθ + θ + θ dv dθ + dv dθ + n V = 4t n V = ξ t n V = θ n V = 6 θ Equation 6 amounts to a weak fmulation of the iginal EPD, its solution will therefe be a weak solution. we notice that 6 is Bessel differential equation of der n whose solution is given by V θ =AJ n θ+byn θ 7 where A and B are constants and J n θ and Y n θ are Bessel functions of der n of first and second kind respectively. Now function Y n θ is singular at the igin i.e. Y n θ as θ. Hence we choose B =so that Now in general, J n ξt is given by V ξ t =AJn ξ t 8 J n ξt= r= r Γr + Γn + r + r+n ξt 9 from which we find the series expansion of J n ξ t as J n ξ t = Γn + J n ξ t = Γ n n ξ t ξ t n Γn + Γ n+ n+ ξ t + n+4 ξ t...! Γn +3 ξ t n+ + ξ t! Γ n+4 n+6...

5 Weak solution of singular Cauchy problem 39 Thus n F[u] =t n V θ =At { F[u]ξ,t=A Γ n Then as t we have that { F[u]ξ, = A { ξ t Γ n n ξ Γ n ξ n Γ n+ Γ n+ ξ t n+ n+ } ξ t +... n } = F ξ by +... } Thus Hence from 5, A = F ξ Γ n ξ n n F[u]ξ,t=t ξ n F ξ Γ n J n ξ t F[u]ξ,t= n n Γ n F ξ ξ t J n ξ t Definition. The inverse Fourier transfm of F ξ inr n denoted F is defined by fx =F [F ξ] = F ξe iξx dξ n The solution of will involve taking the inverse Fourier transfm of both sides and in particular the inverse Fourier transfm of the quantity n ξ t J n ξ t since f F[fx] = F ξ. We state the following theem without proof, f details see [9] Theem 3. Bessel function of der n of first kind may be written in trigonometric fm as ξt n J n ξt= n Γn + cosξtcos θ sin n θdθ

6 3 A. Manyonge, D. Kweyu, J. Bitok, H. Nyambane and J. Maremwa Hence from the theem we have that n ξ t J n ξ t = In this equation, let z = ξ t and n F[Q] =z J n z = 4 n Γ n 4 n Γ n cos ξ t cos θ sin n θdθ cosz cos θ sin n θdθ Take the inverse Fourier transfm of f x R on both sides to find Qx, t = F[Q]e iξx dξ Let Qx, t = Qx, t = Qx, t = so that Qx, t = Qx, t = n 3 Γ n n 3 Γ n Qx, t = n Γ n n 3 Γ n 4 n Γ n cosz cos θ sin n θe iξx dθdξ cosz cos θ sin n θe iξx dθdξ e iz cos θ+ξx + e e i ξ t cos θ+xξ + e iz cos θ ξx E = ei ξ t cos θ+xξ i ξ t cos θ xξ + e n 3 Γ n Edξ + e it cos θ+xξ dξ + i ξ t cos θ xξ sin n θdθdξ sin n θdθdξ 3 Edξ sin n θdθ 4 e it cos θ xξ dξ sin n θdθ

7 Weak solution of singular Cauchy problem 3 Qx, t = n Γ n i t cos θ sin n θdθ 5 t cos θ x We now construct the weak solution of the EPD f n = 4. 5 now becomes Qx, t = 3 t cos θ sin θ Γ 3i dθ t cos θ x Qx, t = 3 3 i t cos θ sin θ t cos θ x dθ 6 To evaluate the integral in 6 we transfm as follows: Let z = e i θ where θ. This means that dθ = dz iz, sinθ = z sin from which we find, cos θ = zθ + iz z, cos θ = z + z t cos θ sin θ = tz + z and t cos θ x = t z + 4x z 8z 3 4z inserting these quantities in the integral we find t cos θ sin θdθ = z + z dz t cos θ x it c z z 4 + 4x z t + where c is a unit circle with centre at the igin. The denominat of the right hand side may be written as t cos θ t cos θ x sin θdθ = it c t z + z dz z z αz + αz βz + β where {x t +x } x t α =, {x t α = +x } x t β = {x t x x t t }, {x t β = x } x t t t

8 3 A. Manyonge, D. Kweyu, J. Bitok, H. Nyambane and J. Maremwa are the roots of the equation z 4 + 4x t z + z = obviously lies within c, f the roots β and β f x>t, it can be shown that they lie in c. Let fz = z + z dz z z 4 + 4x t z + We examine the residues of fz within c. F residue of fz atz =, this is a pole of der given by lim z { d! dz With a little simplification, we find z z + z z z αz + αz βz + β } Residue of fz at z= = α α β 7 F residue of fz atz = β we have lim z β z + z z z αz + αz + β = β + β β ββ α 8 and f residue of fz atz = β we have lim z β z + z z z αz + αz + β = β β β ββ α 9 If ΣR denotes sum of residues of fz inc, then from 7, 8 and 9, ΣR = α α β and from Cauchy integral theem t cos θ sin θdθ = α t cos θ x it i α β t cos θ sin θdθ = α t cos θ x α βt

9 Weak solution of singular Cauchy problem 33 Now our iginal integral occupied only of c, hence we have 4 t cos θ sin θdθ = α t cos θ x α βt. 4 Using in 6 we find Qx, t = t cos θ t cos θ x Qx, t = sin θdθ = 5 iαβt α βtα α 5 i {x t +x x t } Definition 4. F x R, let fx and gx be two given functions. The convolution of fx and gx denoted fx gx is defined to be the function f gx = fygx ydy R n Theem 5 Convolution theem. Let F [F ξ] = fx and F [F ξ] = gx, then F [F ξ F ξ] = fx gx In, set n = 4. Taking the inverse Fourier transfm of this equation and using, becomes ux, t =4fx Qx, t 3 Using the convolution theem in 3, we find ux, t = fydy i {x y t +x y 4 x y t }. Weak solution f n=, the case of -D wave equation Consider when n =, i.e. F[u]x, t = F ξ ξ t J ξ t 5

10 34 A. Manyonge, D. Kweyu, J. Bitok, H. Nyambane and J. Maremwa and becomes ξ t J ξ t = accding to [9]. Using this in 5 we find F[u]x, t =F ξ cos ξ t cos ξ t 6 ux, t = t i fx t x 7 on use of convolution theem. Hence ux, t = t i fydy [t x y ] 8 being the weak solution of the D wave equation. 3 Conclusion Equations 4 and 8 obtained represent the weak solutions of the EPD f n = 4 and n = the D wave equations respectively. References [] Euler, L.: Foundations of Integral Calculus, vol. 3, Academia Scientiarum Imperialis Petropolitanae, 77. [] Martin, M.95: Riemann s method and the Cauchy Problem Bull. Amer. Math. Soc., vol. 57, pp [3] Diaz, J. and Weinberger, H.953: A Solution of the singular INtial Value Problem f the EPD Equation Proc. Amer. Math. Soc., vol. 4, pp [4] Blum, E.954: The Euler-poisson-darboux equation in exceptional Cases Pro. Amer. Math. Soc., vol. 5, pp5-5.

11 Weak solution of singular Cauchy problem 35 [5] Weinstein, A.954: On the Wave Equation and the Euler-Poisson- Darboux Equation Pro. Fifth Symp. App. Math; Amer. Math. Soc., pp [6] Fusaro, B.966: A Solution of a singular Mixed Problem f the Equation of Euler-Poisson-Darboux Amer. Math. Monthly, vol. 8, pp97-9. [7] Dernek, N.: On the Singular Cauchy Problem of Euler-Poisson- Darboux Equation Istanbul Univ. Fen. Fak. Mat. Der.,vol. 59, pp7-7. [8] Uchikoshi, K.7: Singular Cauchy Problems f Perfect incompressible Fluids Jour. Math. Univ. Tokyo., vol. 4 pp [9] Carroll, R. W. and Showalter, R.: Singular and Degenerate Cauchy Problems, vol., New Yk, Academic Press, 976. Received: September,