AN APPROXIMATE ANALYTIC SOLUTION OF A PARTICULAR BOUNDARY VALUE PROBLEM

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the whole x′y′-plane and finite in z′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution. 2000 Mathematics Subject Classification. 35C05, 41A46, 44A15. 1. Mathematical formulation and analysis. The governing equation for the nondimensional temperature distribution is given by [2] Txx+Tyy+Tzz =−(Pe)Tx (1.1) The boundary conditions for the temperature are T → 0 as x →±∞, y →±∞, Tz|z=0 =− 2AP πr0kw ( Tm−T0 ) exp[−2(x2+y2)]+NutT , Tz|z=d =−NubT , (1.2) where A, P , r0, kw , Tm, T0, Pe, Nut , and Nub are all constants defined in [2]. Equation (1.1) can be simplified by defining T = T1 exp ( − Pe 2 x ) (1.3) which transforms (1.1) into the following form: T1xx+T1yy+T1zz− ( Pe2 4 ) T1 = 0 (1.4) and, the boundary conditions are now expressed as T1 → 0 as x →±∞, y →±∞, T1z|z=0 =− 2AP πr0kw ( Tm−T0 ) exp[−2(x2+y2)]+NutT1, T1z|z=d =−NubT1. (1.5) 514 U. TANRIVER AND A. KAR The solution to the boundary value problem was obtained earlier [2]. The solution is given in dimensional form T ′ ( x′,y ′,z′ )= T0+ AP exp [−Pe/2(x−Pe/16)] 2πr0kw ∫∞ 0 Rexp ( − R 2 8 ) Φ(R)J0(rR)dR, (1.6) where J0(β) is the Bessel function of order zero, x = x′/r0, y =y ′/r0, z = z′/r0, Φ(R)= M cosh [ M(d−z)]+Nub sinh[M(d−z)] ( M2+NutNub ) sinh(Md)+M(Nut+Nub)cosh(Md) , M = √ R2+ Pe 2 4 , (1.7) and, R is the distance in the Fourier space (dual variable) defined as R = √ p2+q2 and r is the distance in the real xy-plane defined as r = √ x2+y2. To approximate the integral in (1.6), consider the special case Nub =Nut = 0. For this case (1.7) becomes Φ(R)= cosh [ M(d−z)] M sinh(Md) . (1.8) For large values of M which is R→∞, (1.8) can be approximated as Φ(R)≈ exp(−Mz)+exp [−M(2d−z)] M . (1.9) Therefore, the integral in (1.6) becomes I ≈ ∫∞ 0 Rexp ( − R 2 8 ) exp(−Mz)+exp[−M(2d−z)] M J0(rR)dR. (1.10) Approximation of the following type of integral is necessary in general: IA = ∫∞ 0 Rexp (−aR2)exp (−c√R2+b2 ) √ R2+b2 J0(rR)dR, (1.11) where a, b, and c are positive constants. One can rewrite (1.11) as IA = exp ( ab )∫∞ 0 Rexp (−a√R2+b2√R2+b2 )exp (−c√R2+b2 ) √ R2+b2 J0(rR)dR. (1.12) Expanding one of the repeated terms under the square root sign in the first exponential term in a Taylor series about R = 0 and retaining the first term, one obtains IA ≈ exp ( ab )∫∞ 0 R exp [−(c+ab)√R2+b2] √ R2+b2 J0(rR)dR. (1.13) AN APPROXIMATE ANALYTIC SOLUTION . . . 515 Using the Hankel transform [1], (1.13) can be evaluated as IA = exp ( ab) exp [−b√r 2+(c+ab)2] √ r 2+(c+ab)2 . (1.14) Now the approximated dimensional temperature distribution can be obtained from (1.14) T ′ ( x′,y ′,z′ )≈ T0+ AP exp [−Pe/2(x−Pe/16)] 2πr0kw exp ( Pe2 32 ) ×   ( −Pe/2 √ r 2+(z+Pe/16)2) √ r 2+(z+Pe/16)2 + exp ( −Pe/2 √ r 2+(2d−z+Pe/16)2) √ r 2+(2d−z+Pe/16)2  . (1.15) 2. Results and discussion. Figures 2.1, 2.2, and 2.3 show the numerical and approximate temperature distributions along the z′-axis for various values of the radial distance in the real physical space such as, r = 0, 0.45 and 0.75 mm with laser power 50 W, absorptivity 36%, laser beam radius 0.15 mm and scanning speed 10 mm/s. As expected, the temperature profiles below z′ = 50 μm are not in good agreement for the case of r = 0 because the approximation is accurate for larger values of r . Figure 2.1 depicts that the two temperature values get closer to each other for z′ values lying in the interval [50,200]. As shown in Figures 2.2 and 2.4, the difference between the temperature values decreases along z′-axis for larger values of r . Figure 2.4 shows the comparison between the numerical and approximate temperature values along the y ′-axis for x′ = 0,0.15 mm. For y ′ values up to 80 μm, there is a discrepancy between the two temperature values for x′ = 0. However, it is good for x′ = 0.15 mm. In addition, all the temperature values are in very good agreement for y ′ values lying in the interval [160,450]. The approximate temperature values can also be improved for small and large values of r . One of the expression under the square root sign of exponent is approximated as √ R2+b2 ≈ b +A ′ r +B′ , (2.1) where A′ and B′ are parameters to be determined numerically. In addition, a multiplicative factor C′ is introduced for integral in (1.12). In this case the result is very much improved for r = 0 as shown in Figure 2.5. As r increases, the difference between the numerical and approximate temperature values becomes smaller. These cases are depicted in Figures 2.6 and 2.7. Figure 2.8 shows very good agreement between the two temperatures for x′ = 0 and all y ′ values compared to Figure 2.4. 516 U. TANRIVER AND A. KAR 0 50 100 150 200 0 500

1. Mathematical formulation and analysis.The governing equation for the nondimensional temperature distribution is given by [2] T xx + T yy + T zz = −(P e)T x (1.1)The boundary conditions for the temperature are T → 0 as x → ±∞, y → ±∞, where A, P , r 0 , k w , T m , T 0 , P e, Nu t , and Nu b are all constants defined in [2].Equation (1.1) can be simplified by defining which transforms (1.1) into the following form: and, the boundary conditions are now expressed as (1.5) The solution to the boundary value problem was obtained earlier [2].The solution is given in dimensional form (1.6) where J 0 (β) is the Bessel function of order zero, x = x /r 0 , y = y /r 0 , z = z /r 0 , and, R is the distance in the Fourier space (dual variable) defined as R = p 2 + q 2 and r is the distance in the real xy-plane defined as r = x 2 + y 2 .
To approximate the integral in (1.6), consider the special case Nu b = Nu t = 0.For this case (1.7) becomes For large values of M which is R → ∞, (1.8) can be approximated as (1.9) Therefore, the integral in (1.6) becomes Approximation of the following type of integral is necessary in general: where a, b, and c are positive constants.One can rewrite (1.11) as Expanding one of the repeated terms under the square root sign in the first exponential term in a Taylor series about R = 0 and retaining the first term, one obtains Using the Hankel transform [1], (1.13) can be evaluated as show the numerical and approximate temperature distributions along the z -axis for various values of the radial distance in the real physical space such as, r = 0, 0.45 and 0.75 mm with laser power 50 W, absorptivity 36%, laser beam radius 0.15 mm and scanning speed 10 mm/s.As expected, the temperature profiles below z = 50 µm are not in good agreement for the case of r = 0 because the approximation is accurate for larger values of r .2.4, the difference between the temperature values decreases along z -axis for larger values of r .Figure 2.4 shows the comparison between the numerical and approximate temperature values along the y -axis for x = 0, 0.15 mm.For y values up to 80 µm, there is a discrepancy between the two temperature values for x = 0.However, it is good for x = 0.15 mm.In addition, all the temperature values are in very good agreement for y values lying in the interval [160,450].
The approximate temperature values can also be improved for small and large values of r .One of the expression under the square root sign of exponent is approximated as where A and B are parameters to be determined numerically.In addition, a multiplicative factor C is introduced for integral in (1.12).In this case the result is very much improved for r = 0 as shown in Figure 2.5.As r increases, the difference between the numerical and approximate temperature values becomes smaller.These cases are depicted in Figures 2.6 and 2.7.

Conclusion.
The numerical and approximate temperature values are presented for different values of the radial distance r in the real physical space.The approximation is based on the fact that the integral expression, (1.13), is accurate around R = 0 which means larger values of r .To fix this problem, (2.1) is used for the optimal choices of the parameters A = 12, B = 1, and C = 1.2.This approximation is also good for smaller values of R. The values of these parameters are determined numerically.
Figure 2.1 depicts that the two temperature values get closer to each other for z values lying in the interval [50, 200].As shown in Figures 2.2 and

Figure 2 .
8 shows very good agreement between the two temperatures for x = 0 and all y values compared to Figure2.4.