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The Comparison Study of Hybrid Method with RDTM for Solving Rosenau-Hyman Equation

Derya Arslan
Derya Arslan
   | 31 mar 2020
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Publicado en línea: 31 mar 2020
Páginas: 267 - 274
Recibido: 27 mar 2019
Aceptado: 16 jun 2019
DOI: https://doi.org/10.2478/amns.2020.1.00024
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© 2020 Derya Arslan, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

2D curves of Hybrid method, Exact and RDTM solutions for t = 0,0001.
2D curves of Hybrid method, Exact and RDTM solutions for t = 0,0001.

Fig. 2

3D curves of Hybrid method, Exact and RDTM solutions for t = 0,0001, respectively.
3D curves of Hybrid method, Exact and RDTM solutions for t = 0,0001, respectively.

Fig. 3

The Comparison of Exact, Hybrid Method and RDTM for t = 0,0001.
The Comparison of Exact, Hybrid Method and RDTM for t = 0,0001.

Some properties of the RDTM.

FunctionTransform
u(x,t)Uk(x)=1k ![∂ku(x,t)∂tk]t=0[U_k (x) = \frac{1}{{k{\kern 1pt} !}}\left[ {\frac{{\partial ^k u(x,t)}}{{\partial t^k }}} \right]_{t = 0}
w(x,t) = u(x,t) ± v(x,t)Wk(x) = Uk(x) ±Vk(x)
w(x,t) = αu(x,t),α is a constant.Wk(x) = αUk(x)
w(x,t) = u(x,t)v(x,t)Wk(x)=∑r=0kUr(x)Vk−r(x)[W_k (x) = \sum\nolimits_{r = 0}^k U_r (x)V_{k - r} (x)
w(x,t)=∂u(x,t)∂t[w(x,t) = \frac{{\partial u(x,t)}}{{\partial t}}Wk(x) = (k + 1 )Uk+1(x)

Comparison of the approximate solutions (hybrid method and RDTM) with exact solution u(x,t) and the error values for t = 0.0001.

x(xi)Hybrid MethodRDTMExactError of RDTMError of Hybrid Method
0.0−2.666917024−2.666666656−2.6666666650.9x10−80.000250359
0.1−2.665253753−2.664992564−2.6650036780.0000111140.000250075
0.2−2.660261708−2.659990035−2.6600122080.0000221730.000249500
0.3−2.651953366−2.651671588−2.6517047320.0000331440.000248635
0.4−2.640349496−2.640058033−2.6401020140.0000439810.000247483
0.5−2.625479097−2.625178409−2.6252330550.0000546460.000246043
0.6−2.607379341−2.607069925−2.6071350190.0000650940.000244322
0.7−2.586095466−2.585777858−2.5858531420.0000752840.000242324
0.8−2.561680672−2.561355438−2.5614406190.0000851810.000240054
0.9−2.534195983−2.533863725−2.5339584660.0000947410.000237517
1.0−2.503710096−2.503371438−2.5034753760.0001039380.000234720

Some properties of differential transform based on t – variable.

FunctionTransform
dw(x,t)dt[\frac{{dw(x,t)}}{{dt}}W (i,k) = (k + 1 )W (i,k + 1 )
w(x,t) = αw(x,t)W (i,k) = αW (i,k)

Some properties of central difference and differential transform based on x–variable.

FunctionTransform
∂w(x,t)∂x[\frac{{\partial w(x,t)}}{{\partial x}}W(i,k)=W(i+1,k)−W(i−1,k)2h[W(i,k) = \frac{{W(i + 1,k) - W(i - 1,k)}}{{2h}}
∂2w(x,t)∂x2[\frac{{\partial ^2 w(x,t)}}{{\partial x^2 }}W(i,k)=W(i+1,k)−2W(i,k)+W(i−1,k)h2[W(i,k) = \frac{{W(i + 1,k) - 2W(i,k) + W(i - 1,k)}}{{h^2 }}
∂3w(x,t)∂x3[\frac{{\partial ^3 w(x,t)}}{{\partial x^3 }}W(i,k)=W(i+2,k)−2W(i+1,k)+2W(i−1,k)−W(i−2,k)2h3[W(i,k) = \frac{{W(i + 2,k) - 2W(i + 1,k) + 2W(i - 1,k) - W(i - 2,k)}}{{2h^3 }}
u(x,t)∂3w(x,t)∂x3[u(x,t)\frac{{\partial ^3 w(x,t)}}{{\partial x^3 }}W(i,k)=∑m=0kW(i,k−m)W(i+2,k)−2W(i+1,k)+2W(i−1,k)−W(i−2,k)2h3[W(i,k) = \sum\nolimits_{m = 0}^k W(i,k - m)\frac{{W(i + 2,k) - 2W(i + 1,k) + 2W(i - 1,k) - W(i - 2,k)}}{{2h^3 }}
∂w(x,t)∂x∂x2w(x,t)∂x2[\frac{{\partial w(x,t)}}{{\partial x}}\frac{{\partial x^2 w(x,t)}}{{\partial x^2 }}W(i,k)=∑m=0kW(i+1,k)−2W(i,k)+W(i−1,k)h2W(i+1,k−m)−W(i−1,k−m)2h[W(i,k) = \sum\nolimits_{m = 0}^k \frac{{W(i + 1,k) - 2W(i,k) + W(i - 1,k)}}{{h^2 }}\frac{{W(i + 1,k - m) - W(i - 1,k - m)}}{{2h}}
u(x,t)∂w(x,t)∂x[u(x,t)\frac{{\partial w(x,t)}}{{\partial x}}W(i,k)=∑m=0kU(i,k−m)W(i+1,k)−W(i−1,k)2h[W(i,k) = \sum\nolimits_{m = 0}^k U(i,k - m)\frac{{W(i + 1,k) - W(i - 1,k)}}{{2h}}
w(x,t) = sinh(x)W (x,t) = sinh(xi)
w(x,t) = cosh(x)W (x,t) = cosh(xi)
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