Vol.:(0123456789) 1 3
Engineering with Computers
https://doi.org/10.1007/s00366-020-01037-4
ORIGINAL ARTICLE
Error analysis and numerical solution of Burgers–Huxley equation
using 3‑scale Haar wavelets
Shitesh Shukla
1
· Manoj Kumar
1
Received: 25 February 2020 / Accepted: 2 May 2020
© Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract
In this article, we develop an efcient and accurate numerical scheme based on the Crank–Nicolson fnite diference
method and Haar wavelet analysis to evaluate the numerical solution of the Burgers–Huxley equation. The present method
is extended form of Haar wavelet 2D scaling which shows that it is reliable for solving nonlinear partial diferential
equations. The numerical results are more accurate than other existing methods available in the literature and very close
to the exact solution. The Haar basis function is generated from multi-resolution analysis and used to evaluate fast and
accurate approximate solutions on the collocation points. The convergence of the proposed method is demonstrated by
its error analysis. We compared numerical solutions with the exact solutions and solutions available in the literature. The
proposed method is found to be straight forward, accurate with small computational cost and can be easily implemented
in mathematical software MATLAB.
Keywords Haar operational matrix · 3D scaling · Quasilinearization technique · Piecewise continuous function ·
Collocation points
1 Introduction
In this article, a modifed numerical algorithm based on
3-scale Haar wavelets is proposed to obtain the approximate
solution of Burgers–Huxley partial diferential equations of
the following form:
with boundary conditions
where
The exact solution [1] of Eq. (1) is
Burgers–Huxley equation is a combination of Burgers and
Huxley equation. Burgers and Huxley equation can be gen-
erated from Eq. (1) for a particular value of and . When
= 0 and = 1, it reduces Huxley equation; when = 0
and = 1, it reduces Burgers equation [2]; when = 0 ,
=-1 and = 1, it reduces Newell–Whitehead equation
[3]. Huxley equation describes the nerve propagation in
environmental science. We can evaluate molecular CB
(1) u
t
+ u
u
x
- u
xx
= u(1 - u
)(u
- )
(2) u(x,0)=
[
2
+
2
tanh(A
1
x)
]
1
(3) u(0, t)=
[
2
+
2
tanh(-A
1
A
2
t)
]
1
, t ≥ 0
(4) u(1, t)=
[
2
+
2
tanh(A
1
(1 - A
2
t))
]
1
, t ≥ 0
(5) A
1
=
- +
√
2
+ 4 (1 + )
4(1 + )
(6)
A
2
=
(1 + )
-
(1 + - )
�
- +
√
2
+ 4 (1 + )
�
2(1 + )
.
(7) u(x, t)=
[
2
+
2
tanh(A
1
(x - A
2
t))
]
1
, t≥0
.
* Shitesh Shukla
shitesh@mnnit.ac.in
Manoj Kumar
manoj@mnnit.ac.in
1
Department of Mathematics, Motilal Nehru National
Institute of Technology Allahabad, Prayagraj, UP 211004,
India