AN INTEGRAL TRANSFORM APPLIED TO SOLVE THE STEADY HEAT TRANSFER PROBLEM IN THE HALF-PLANE

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1 THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S S05 AN INTEGRAL TRANSFORM APPLIED TO SOLVE THE STEADY HEAT TRANSFER PROBLEM IN THE HALF-PLANE by Tongqiang XIA a, Shengping YAN b, Xin LIANG b, Pengjun ZHANG a, and Chun LIU c * a School of Electric and Power Engineering, China University of Mining and Technology, Xuzhou, China b School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, China c IoT Perception Mine Research Center, China University of Mining and Technology, Xuzhou, China Original scientific paper An integral transform operator it A Π Π is considered to solve the steady heat transfer problem in this paper. The analytic technique is illustrated to be applicable in the solution of a -D Laplace equation in the half-plane. The results are interesting as well as potentially useful in the linear heat transfer problems. Key words: heat transfer, Laplace equation, analytic solution, integral transform Introduction The PDE were used to describe the heat transfer problems, such as diffusion [-4], and Laplace equation [5-7]. The PDE in the steady heat transfer problems [5-9] are a second-order PDE named after Pierre-Simon Laplace who first researched its properties. Its extended versions were discussed by different researcher in the sense of the fractional- and fractal-order space operators. For example, the fractional Laplacian equations with the critical Sobolev exponent were considered in [0]. The weak and numerical solutions were reported in [, ]. The local fractional Laplace equation with local fractional derivative was reported in [3, 4]. The (linear) Laplace equations in the different conditions were solved by the integral transforms [5]. A novel Fourier-like integral transform operator was proposed to find the exact solutions for a steady heat transfer problem [6]. The Fourier-like integral transform technique for the Laplace equation has not yet considered. The brief target of the manuscript is to consider the proposed method for find the analytic solution for the Laplace equation. A novel integral transform technique In this paper, we present the properties of the novel integral transform operator which was proposed in [6] and used in the paper. * Corresponding author, lccumt@63.com

2 S06 Xia, T., et al.: An Integral Transform Applied to Solve the Steady Heat Transfer Problem... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S The integral transform of the function Π () t is [6]: it Π ( ) A Π Π () where A is the Fourier-like integral transform operator. The inverse Fourier-like integral transform operator is defined [6]: it Π A Π Π e d () provided the integral exists for some. As a direct result of eqs. () and (), an integral formula is presented [6]: it it Π A Π Π e d (3) The properties of the Fourier-like integral transform operator are [6]. (U) Suppose that Π ( ω) [ Π] and Π ( κ) A [ Π]. Then we have: at (U) If Π () t e ut (), then we have: (U3) If Π () t δ () t, then we have: (U4) Let Π A [ Π] (U5) Let Π A [ Π] Π ( ω) Π and ( ω) Π ( ) ( a+ i) Π Π (4) ω (5) Π ( ω) (6) e i κ t a a ( κ ) A Π Π (7) Π A i Π (8) dt As the direct results, we have: Π A Π (9) dt 3 Π 3 A i 3 Π (0) dt Π 4 A 4 4 Π () dt

3 THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S S07 (U6) Let Π A [ Π] and Θ A [ Θ] n Π n A n ( i) Π () dt (U7) Let Π A [ Π] and Θ A [ Θ] (U7) Let Π A [ Π] A Π +Θ Π +Θ (3) A Π( t τ) Θ ( τ) dτ Π Θ (4) A Π( t ν) Θ ( ν) dν Π Θ (5) The properties of the Fourier-like integral transform operator are presented. (S) If Π () t t n e iat, then we have: i n n ( a) Π δ (6) where δ( ) is the Dirac function [5]. Proof. According to the definition of the Fourier-like integral transform operator, we have: n n iat it i Π A Π t e δ a where the formula [5]: ( n ) n iatx it n ( n x e i δ ) ( a) is valid. (S) If Π sgn, where sgn( t ) is the sign function [5], then we have: Π ( ) (7) i Proof. Using the definition of the Fourier-like integral transform operator, we obtain: where there is [5]: it Π ( ) A Π sgn i it sgn i

4 S08 Xia, T., et al.: An Integral Transform Applied to Solve the Steady Heat Transfer Problem... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S (S3) If Π /( t + a ), where a > 0, then we have: e a Π (8) a Proof. Considering the following relationship [5]: we have: a it t + a a e (S4) If e a it Π A Π t t + a a Π () t e at, where a > 0, then we have: 4 a Π ( ) e a (9) Proof. Using the following relationship [5]: we get: a at it 4a e e at i t 4 a Π ( ) A Π e e a (S5) If Π () t e at, where a > 0, then we have: ( ) Π Proof. With use of the relationship [5]: a ( a + ) (0) we have: e a at i t a + e at e i t dt a Π A Π (S6) If Π () t e iat, where a > 0, then we have: Proof. We have [5]: ( ) ( a + ) δ a Π ( ) () iat it e δ ( a)

5 THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S S09 ( ) iat it δ a Π ( ) A Π e (S7) If Π sin t, then we have: Proof. We have: δ ( ) δ + Π ( ) () i it it it e e it sint δ ( ) δ ( + ) i i δ δ + it Π ( ) A sin te dt Π i (S8) If Π cost, then we have: Proof. We have: δ ( ) δ + + Π ( ) (3) it it it e + e it cost δ ( ) + δ ( + ) δ + δ + it Π ( ) A Π coste dt Solving the -D Laplace equation In this section, we handle a Dirichlet s problem for a -D Laplace equation in the half-plane. Let us consider a -D Laplace equation in the half-plane [5]: Θ x with the boundary conditions: ( xy, ) Θ( xy, ) + 0, < x<, y 0 y ( xy, ) 0 (4) Θ x,0 g x, < x< (5) Θ as x, y (6) Introducing the Fourier-like integral transform operator with respect to x, we have: ix Θ (, y) A Θ ( xy, ) Θ( xy, ) e dx (7)

6 S0 Xia, T., et al.: An Integral Transform Applied to Solve the Steady Heat Transfer Problem... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S which leads to: where and Θ (, y) y ( y) Θ (,0) g, 0 (8) Θ (9) ( y) Θ, 0 as y (30) Thus, finding the solution of eq. (7), we obtain from eqs. (8) and (9) that: ( y) g Θ, y e (3) Thus, we have: y y A e x + y (3) y Θ (, y) g e (33) Thus, we have from eq. (33): y g ( ν) dν Θ ( xy, ), x ν + y y 0 (34) which is in agreement with the result from Fourier transform [5]. Conclusion In the present work, a novel Fourier-like integral transform technique was used to solve the PDE in heat transfer. The analytical solution of the Laplace equation was obtained. Comparing with the result from Fourier transform, the presented method is efficient, accurate and alternative for finding the linear PDE from the problems in science and engineering. Acknowledgment This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (IRT7R03), the creative research groups of China (54003), the National Natural Science Funds of China ( and 55045) and the Fundamental Research Funds for the Central Universities (04XT0). This work was also a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. Nomenclature t temperature, [K] x, y space co-ordinates, [m] References [] Pletcher, R. H., et al., Computational Fluid Mechanics and Heat Transfer, CRC Press, Boca Raton, Fla., USA, 0

7 THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S05-S S [] Yang, X. J., A New Integral Transform Operator for Solving the Heat-Diffusion Problem, Applied Mathematics Letters, 64 (07), Feb., pp [3] Yang, X. J., Gao, F., A New Technology for Solving Diffusion and Heat Equations, Thermal Science, (07), A, pp [4] Liang, X, et al., Applications of a Novel Integral Transform to Partial Differential Equations, Journal of Nonlinear Science and Applications, 0 (07),, pp [5] Sommerfeld, A., Partial Differential Equations in Physics, Academic Press, New York, USA, 949 [6] Sangani, A. S., Acrivos, A., Slow Flow Past Periodic Arrays of Cylinders with Application to Heat Transfer, International Journal of Multiphase Flow, 8 (98), 3, pp [7] Domenico, P. A., Palciauskas, V. V., Theoretical Analysis of Forced Convective Heat Transfer in Regional Ground-Water Flow, Geological Society of America Bulletin, 84 (973),, pp [8] Yang, X. J., A New Integral Transform Method for Solving Steady Heat Transfer Problem, Thermal Science, 0 (06), Suppl. 3, pp. S639-S64 [9] Yang, X. J., A New Integral Transform with an Application in Heat Transfer Problem, Thermal Science, 0 (06), Suppl. 3, pp. S677-S68 [0] Servadei, R., Valdinoci, E., Fractional Laplacian Equations with Critical Sobolev Exponent, Revista Matemática Complutense, 8 (05), 3, pp [] Servadei, R., Valdinoci, E., Weak and Viscosity Solutions of the Fractional Laplace Equation, Publicacions Matematiques, 58 (04),, pp [] Acosta, G., Borthagaray, J. P., A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations, SIAM Journal on Numerical Analysis, 55 (07),, pp [3] Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Academic Press, New York, USA, 05 [4] Yang, X. J., et al., A New Family of the Local Fractional PDE, Fundamenta Informaticae, 5 (07), -4, pp [5] Debnath, L., Bhatta, D., Integral Transforms and their Applications, CRC Press, Boca Raton, Fla., USA, 04 [6] Yang, X. J., A New Integral Transform Method for Soliving a Steady Heat Transfer Problem, Thermal Science, 0 (06), Suppl., pp. S639-S64 Paper submitted: March 0, 07 Paper revised: May, 07 Paper accepted: June 7, Society of Thermal Engineers of Serbia Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditions