08092017  Issue 3/2018 Open Access
A New Functional Iterative Algorithm for the Regularized LongWave Equation Using an Integral Equation Formalism
 Journal:
 Journal of Scientific Computing > Issue 3/2018
1 Introduction
2 Integral Formalism
2.1 Integral Transformations
2.2 Inverse Integral Transformations
2.3 Establishing Integral Formalism
3 Derivation of Functional Iteration Algorithm
3.1 Dual Integral Equations
3.2 Deriving Functional Iterative Algorithm
3.3 Iterative Algorithm for the Special Case \(\alpha =0\)
4 Numerical Experiments
4.1 Numerical Performance
4.1.1 Simulations for \(\alpha =0\)
Time \(t^{*}\)

\(Err_{2}\times 10^{3}\)

\(Err_{\infty }\times 10^{3}\)

\(C_{1}\)

\(C_{2}\)

\(C_{3}\)


Analytical  0  0  3.97993  0.81046  2.57901 
0  0  0  3.97993  0.81046  2.57901 
4  0.007657  0.005538  3.97993  0.81046  2.57901 
8  0.008472  0.003726  3.97992  0.81046  2.57901 
12  0.008711  0.003367  3.97993  0.81046  2.57901 
16  0.008779  0.003102  3.97992  0.81046  2.57901 
20  0.008929  0.002902  3.97988  0.81046  2.57901 
The proposed 
\(Err_{2}\times 10^{3}\)

\(Err_{\infty }\times 10^{3}\)

\(C_{1}\)

\(C_{2}\)

\(C_{3}\)


\(\Delta x^{*}=\Delta t^{*}=0.50\) (\(\hbox {M}=201, \hbox {N}=41\))  0.096148  0.038098  3.97989  0.81046  2.57901 
\(\Delta x^{*}=\Delta t^{*}=0.25\)
\((\hbox {M}=401, \hbox {N}=81)\)
 0.025281  0.009571  3.97988  0.81046  2.57901 
\(\Delta x^{*}=1/8,\Delta t^{*}=0.1\)
\((\hbox {M}=801, \hbox {N}=201)\)
 0.008929  0.002902  3.97988  0.81046  2.57901 
The conventional \(\Delta x^{*}=1/8\), \(\Delta t^{*}=0.1\) \((\hbox {M}=801, \hbox {N}=201)\)
 

P.Q.S. [14]  0.10966  0.04101  3.97989  0.81046  2.57901 
G.M. [15]  0.26659  0.09146  3.97964  0.81026  2.57873 
L.S.L. [16]  4.68800  1.75500  3.98203  0.80865  2.57302 
G.L. [17]  0.51100  0.19800  3.98206  0.81116  2.58133 
C.C. [18]  0.37841  0.13993  3.97995  0.81046  2.57900 
QBGM1 [19]  0.19215  0.07337  3.97988  0.81046  2.57900 
QBGM2 [19]  0.35489  0.12848  3.97988  0.81046  2.57900 
PG FEM [13]  0.06493  0.02643  3.97990  0.81050  2.57900 
SQBS CM [20]  0.04315  0.01321  3.97989  0.81050  2.57900 