# Numerical similarity solution for a variable coefficient K(m, n) equation

## Abstract

A technique for finding numerical similarity solutions to an initial boundary value problem (IBVP) for generalized K(m, n) equations is described. The equation under consideration is nonlinear and has variable coefficients. The original problem is transformed with the aid of Lie symmetries to an initial value problem (IVP) for a nonlinear third-order ordinary differential equation. The existence and uniqueness of the solution are examined, and the problem is consequently solved with the aid of a finite-difference scheme for various values of the governing parameters. In lieu of an exact symbolic solution, the scheme is validated by comparing the numerical solutions with the approximate analytic solutions obtained with the aid of the method of successive approximations in their region of validity. The accuracy, efficiency, and consistency of the scheme are demonstrated. Numerical solutions to the original initial boundary value problem are constructed for selected parameter values with the aid of the transforms. The qualitative behavior of the solutions as a function of the governing parameters is analyzed, and it is found that the examined IBVPs for generalized K(m, n) equations with variable coefficients that are functions of time, do not admit solitary wave or compacton solutions.

- Authors:

- University of Nicosia, Department of Mathematics (Cyprus)
- University of Cyprus, Department of Mathematics and Statistics (Cyprus)

- Publication Date:

- OSTI Identifier:
- 22769357

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ANALYTICAL SOLUTION; APPROXIMATIONS; BOUNDARY-VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; FINITE DIFFERENCE METHOD; NONLINEAR PROBLEMS; SYMMETRY; TIME DEPENDENCE

### Citation Formats

```
Christou, Marios A., E-mail: christou.ma@unic.ac.cy, Papanicolaou, Nectarios C., E-mail: papanicolaou.n@unic.ac.cy, and Sophocleous, Christodoulos.
```*Numerical similarity solution for a variable coefficient K(m, n) equation*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0387-8.

```
Christou, Marios A., E-mail: christou.ma@unic.ac.cy, Papanicolaou, Nectarios C., E-mail: papanicolaou.n@unic.ac.cy, & Sophocleous, Christodoulos.
```*Numerical similarity solution for a variable coefficient K(m, n) equation*. United States. doi:10.1007/S40314-016-0387-8.

```
Christou, Marios A., E-mail: christou.ma@unic.ac.cy, Papanicolaou, Nectarios C., E-mail: papanicolaou.n@unic.ac.cy, and Sophocleous, Christodoulos. Tue .
"Numerical similarity solution for a variable coefficient K(m, n) equation". United States. doi:10.1007/S40314-016-0387-8.
```

```
@article{osti_22769357,
```

title = {Numerical similarity solution for a variable coefficient K(m, n) equation},

author = {Christou, Marios A., E-mail: christou.ma@unic.ac.cy and Papanicolaou, Nectarios C., E-mail: papanicolaou.n@unic.ac.cy and Sophocleous, Christodoulos},

abstractNote = {A technique for finding numerical similarity solutions to an initial boundary value problem (IBVP) for generalized K(m, n) equations is described. The equation under consideration is nonlinear and has variable coefficients. The original problem is transformed with the aid of Lie symmetries to an initial value problem (IVP) for a nonlinear third-order ordinary differential equation. The existence and uniqueness of the solution are examined, and the problem is consequently solved with the aid of a finite-difference scheme for various values of the governing parameters. In lieu of an exact symbolic solution, the scheme is validated by comparing the numerical solutions with the approximate analytic solutions obtained with the aid of the method of successive approximations in their region of validity. The accuracy, efficiency, and consistency of the scheme are demonstrated. Numerical solutions to the original initial boundary value problem are constructed for selected parameter values with the aid of the transforms. The qualitative behavior of the solutions as a function of the governing parameters is analyzed, and it is found that the examined IBVPs for generalized K(m, n) equations with variable coefficients that are functions of time, do not admit solitary wave or compacton solutions.},

doi = {10.1007/S40314-016-0387-8},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}