TY - JOUR

T1 - Numerical similarity solution for a variable coefficient K(m, n) equation

AU - Christou, Marios A.

AU - Papanicolaou, Nectarios C.

AU - Sophocleous, Christodoulos

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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.

AB - 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.

KW - Finite-difference methods

KW - K(m, n) equations

KW - Lie symmetries

KW - Nonlinear differential equations

UR - http://www.scopus.com/inward/record.url?scp=85047435956&partnerID=8YFLogxK

U2 - 10.1007/s40314-016-0387-8

DO - 10.1007/s40314-016-0387-8

M3 - Article

AN - SCOPUS:85047435956

SN - 0101-8205

VL - 37

SP - 1098

EP - 1111

JO - Computational and Applied Mathematics

JF - Computational and Applied Mathematics

IS - 2

ER -