THE FOURIER-YANG INTEGRAL TRANSFORM FOR SOLVING THE 1-D HEAT DIFFUSION EQUATION. Jian-Guo ZHANG a,b *

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1 Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 S63 THE FOURIER-YANG INTEGRAL TRANSFORM FOR SOLVING THE -D HEAT DIFFUSION EQUATION by Jian-Guo ZHANG a,b * a Sae Key Laboraory of Coking Coal Eploiaion and Comprehensive Uilizaion, China Pingmei Shenma Group, Pingdingshan, Henan, China b China Pingmei Shenma Group, Pingdingshan, Henan, China Original scienific paper hps://doi.org/0.98/tsci7s063z A new Fourier-like inegral ransform (called he Fourier-Yang inegral ransform) j ( ) ( )e d Λ = Λ is considered o find he fundamenal soluions of he -D hea diffusion equaion in he differen iniial condiions. Key words: fundamenal soluion, hea equaion, diffusion equaion, Fourier-Yang inegral ransform, Fourier-like inegral ransform Inroducion The PDE in he hea ransfer problems are he imporan opics for scieniss and engineers o eplore he hea ranspor in he solid, liquid and gas [-4]. The hea diffusion equaion is one of he ineresing PDE for describe he hea ransfer heory [5-7] and he diffusion flow in meamorphic rocks [8, 9]. Wih he aid of he (non-local and local) fracional calculus, he hea diffusion equaion can be generalized o fracional diffusion equaions [0-] and local fracional diffusion equaions [3-5]. In order o find he soluions for he hea diffusion equaions, many echnologies, such as he Laplace-like inegral ransform [5], finie inegral ransform [6], homology [7], variaional ieraion [8], alernaing-direcion implici [9], immersed inerface [0], and he Laplace-like inegral ransform [] mehods, were developed. A new Fourier-like inegral ransform (called he Fourier-Yang inegral ransform), proposed by Yang [], was considered o solve he seady hea ransfer problem. More inegral ransforms for solving he hea ransfer problems were considered in [3-5]. The aim of he presen manuscrip is o presen he properies of his inegral ransform and a new applicaion o find he fundamenal soluion for a -D hea diffusion equaion. The Fourier-Yang inegral ransform In his secion, we inroduce he conceps of Fourier and Fourier-Yang inegral ransforms, and properies of he Fourier-Yang inegral ransform. The Fourier inegral ransform of he funcion Φ () is given [3]: Auhorʼs @63.com

2 S64 Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 ( θ ) ( ) ( ) jθ Φ = Φ = : Φ e d () is he Fourier inegral ransform operaor. The inverse Fourier inegral ransform operaor of eq. (4) is wrien [3]: ( ) ( θ) ( ) jθ Φ = Φ = : Φ e dθ π j () is he inverse Fourier inegral ransform operaor. The Fourier inegral formula is given [3]: jθ jθ Φ ( ) = Φ ( θ) = Φ( ) e d e dθ π The new Fourier-Yang inegral ransform of he funcion Λ () is given []: ( ) ( ) ( ) (3) j Λ = Λ = : Λ e d (4) is he new Fourier-Yang inegral ransform operaor. The inverse Fourier-Yang inegral ransform operaor is defined []: ( ) Λ j Λ ( ) = Λ ( ) = : e d π (5) is he inverse Fourier-Yang inegral ransform operaor. The Fourier-Yang inegral formula is given []: j j Λ ( ) = Λ ( ) = : Λ( ) e d e d π (6) Taking ϖ = j, we obain he Laplace-Carson inegral ransform of he funcion Ω () [4]: γ ( γ) ( ) ( ) γ Ω =R Ω = : Π e d (7) R is he Laplace-Carson inegral ransform operaor. Similarly, he inverse Laplace-Carson inegral ransform operaor is presened [4]: 0 ω0 i ( γ ) ω0 + i Ω γ Ω ( ) =R Ω ( γ) = : e dγ π j (8) γ The properies of he Fourier-Yang inegral ransform operaor are as follows []. (T) If Λ ( θ ) = Λ [ ( )] and Λ ( ) = [ Λ( )], hen we have: Λ ( θ) = Λ ( ) and ( θ) ( ) Λ = Λ (9) a (T) If Λ () = e ϕ(), ϕ () is he Heaviside uni sep funcion, hen we have:

3 Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 S65 Λ ( ) = (0) a + j a is a consan. (T3) If Λ () = δ (), δ () represens he Dirac funcion, hen we have: Λ ( ) = () (T4) If Λ ( ) = [ Λ( )], hen we have: ( a) e ja Λ = Λ( ) () (T5) If Λ ( ) = [ Λ( )], hen we have: ( ) dλ = j Λ ( ) (3) d (T6) If Λ ( ) = [ Λ( )], hen we have: d Λ ( ) (T7) If Λ ( ) = [ Λ( )] and Θ ( ) = [ Θ( )], hen we have: = Λ( ) (4) d ( ) ( ) ( ) ( ) Λ +Θ =Λ +Θ (5) (T8) If Λ ( ) = [ Λ( )] and Θ ( ) = [ Θ( )], hen we have: Λ( τ) Θ ( τ) dτ = Λ( ) Θ( ) (6) (T9) If Λ ( ) = [ Λ( )], hen we have: S Λ ( ) d = Λ( ) (7) j (T0) If Λ () = be a, a > 0, hen we have: 4 a bπ Λ ( ) = e (8) π a Proof. We have, by he definiion of he Fourier-Yang inegral ransform, ha: Λ ( ) = be e d = be d = be e d = e πa j a + a 4a a j 4 a a b π 4a (9) a π e d = (0) a

4 S66 (T) If hen we have: (T) If Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 Μ, Λ ( ) = () 0 / j Λ ( ) = Μ e d = Μsin () / hen we have: ( ) Λ = e πζ ζ e ζ ( ) Λ = (3) Proof. By he definiion of he Fourier-Yang inegral ransform we have: ζ ( + jζ ) ζ j ζ Λ ( ) = e e d e d = = ζ π πζ ζ ζ ζ = e e d = e (4) πζ The fundamenal soluion for he -D hea diffusion equaion In his secion, we use he Fourier-Yang inegral ransform o solve a -D hea diffusion equaion wih he differen iniial condiions. We now consider he iniial value problem for a -D hea diffusion equaion wihou source or sinks [3]: Λ(,) Λ(,) = ψ, < <, 0< (5) ψ is he diffusiviy consan wih he iniial condiion: Λ (,0) = g ( ), < < (6) We find he Fourier-Yang inegral ransform for his problem wih respec o he space variable. Le us consider he following equaions: (, ) (, ) (, ) Λ Λ j Λ = e d = (7)

5 Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 S67 Λ (, ) Λ(, ) j e d = = Λ (, ) (8) Subsiuing eqs. (7) and (8) ino eq. (5), we have: ( ) d Λ, d ψ ( ) 0 + Λ =, 0 < (9) Λ (,0) = g( ) (30) Finding he soluion of eq. (7), we have: ψ Λ (, ) = g( )e (3) Making use of he inverse Fourier-Yang inegral ransform, we ge: ψ j j ψ g( )e Λ (,) = [ Λ (,) ] = e d = g()e d π π From eq. (6), we have: which leads o: Λ( τ, ) Θ ( τ, ) d τ = Λ(, ) Θ(, ) (3) Λ(, ) Θ (, ) = Λ( τ, ) Θ( τ, ) dτ (33) In view of eq. (33), we have: Thus, from eq. (3), we obain: Λ (,) = g( τ,) Θ(,)d τ τ (34) ( τ ) ψ, S e Θ = (35) Λ ( ) g( τ) ( τ ), = e dτ (36) This resul is wih agreemen wih he soluion of he -D hea diffusion equaion by using Fourier ransform [3]. Le Λ (,0) = g ( ) = δ ( ) in eq. (6). Then, from eq. (36) we have: ψ Λ (,) = e (37) Wih he use of he inverse Fourier-Yang inegral ransform, we have:

6 S68 Zhang, J.-G., e al.: The Fourier-Yang Inegral Transform for Solving he -D... THERMAL SCIENCE: Year 07, Vol., Suppl., pp. S63-S69 ψ j j ψ e Λ (, ) = Λ (, ) = e d = e d π π (38) Thus, we obain he soluion for he -D hea diffusion equaion: which resuls in: ( τ ) Λ(, )= δτ ( )e dτ (39) Λ(, )= e (40) This resul is in accordance wih he soluion of he -D hea diffusion equaion by using Fourier-like ransform [5]. Le Λ (,0) = g ( ) = e in eq. (6). Then, from eq. (36) we have he soluion in he Fourier-Yang inegral ransform: which leads o: 4 ψ Λ (,) = π e e (4) By he inverse Fourier-Yang inegral ransform, we have: 4 ψ πe e j 4 j ψ ( ) ( ) Λ, = Λ, = e d = πe e d π π (4) ( τ ) Λ(, ) = e e dτ (43) Conclusion We presen he new applicaion of he Fourier-Yang inegral ransform o solve he iniial value problem for he -D hea diffusion equaion in his work. The fundamenal soluions of his problem wih he iniial condiions were obained wih he use of he Fourier-Yang inegral ransform. The approach for solving his problem is efficien and accurae. Nomenclaure ime, [s] space co-ordinae, [m] Greek symbols Λ(,) emperaure, [K] ψ diffusiviy consan, [Wm K ] References [] Bergman, T. L., Inroducion o Hea Transfer, John Wiley and Sons, New York, USA, 0 [] Io, K., Diffusion Processes, John Wiley and Sons, New York, USA, 974 [3] Luikov, A. V., Analyical Hea Diffusion Theory, Elsevier, New York, USA, 0 [4] Shewmon, P., Diffusion in Solids. Springer, New York, USA, 06 [5] Yang, X. J., A New Inegral Transform Operaor for Solving he Hea-Diffusion Problem, Applied Mahemaics Leers, 64 (07), Feb., pp

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