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New insights into singularity analysis

  • Amlan K. Halder ORCID logo EMAIL logo , Andronikos Paliathanasis ORCID logo and Peter G. L. Leach
Published/Copyright: August 9, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 23 Issue 7-8

Abstract

In this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent.

Keywords: closed-form solution; similarity solutions; singularity analysis; symmetry analysis
1991 Mathematics Subject Classification: 34A05; 34A34; 34C14; 22E60; 35B06; 35C05; 35C07
Corresponding author: Amlan K. Halder, Department of Mathematics, Pondicherry University, Puducherry 605014, India, E-mail:

Funding source: Durban University of Technology

Funding source: UGC (India)

Award Identifier / Grant number: F1-17.1/201718/RGNF-2017-18-SC-ORI-39488

Funding source: National Research Foundation of South Africa

Funding source: University of KwaZulu-Natal

Acknowledgments

AKH dedicate this work to Late Prof. K. M.Tamizhmani, on the occasion of his 65th birth anniversary to commemorate him.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: AKH acknowledges the financial support of UGC (India), NFSC, Award No. F1-17.1/201718/RGNF-2017-18-SC-ORI-39488. PGLL acknowledges the support of the National Research Foundation of South Africa, the University of KwaZulu-Natal, and the Durban University of Technology.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Computation of resonances for Eq. (6.5)

The resultant equation for the computation of resonances is

0 = − 14 a 7 ( x − x 0 ) 21 4 − 2187 a 3 μ 1 μ 2 μ 4 4 32 ( x − x 0 ) 21 4 − 243 a 5 μ 1 2 μ 4 2 x 4 ( x − x 0 ) 19 4 + 63 a 6 μ 1 μ 4 ( x − x 0 ) 9 2 − 81 a 5 μ 1 2 μ 4 2 ( x − x 0 ) 15 4 − 98 a 6 m ( x − x 0 ) − 21 4 + s − 6561 a 2 μ 1 μ 2 μ 4 4 m ( x − x 0 ) − 21 4 + s 32 + 9477 a 2 μ 1 μ 2 μ 4 4 m s ( x − x 0 ) − 21 4 + s 8 + 6561 4 a 2 μ 1 μ 2 μ 4 4 m s 2 ( x − x 0 ) − 21 4 + s + 729 a 2 μ 1 μ 2 μ 4 4 m s 3 ( x − x 0 ) − 21 4 + s − 1215 4 a 4 μ 1 2 μ 4 2 m x ( x − x 0 ) − 19 4 + s + 81 a 4 μ 1 2 μ 4 2 m s x ( x − x 0 ) − 19 4 + s + 378 a 5 μ 1 μ 4 m ( x − x 0 ) − 9 2 + s − 405 a 4 μ 1 2 μ 4 2 m ( x − x 0 ) − 15 4 + s − 294 a 5 m 2 ( x − x 0 ) − 21 4 + 2 s − 6561 32 a μ 1 μ 2 μ 4 4 m 2 ( x − x 0 ) − 21 4 + 2 s + 9477 4 a μ 1 μ 2 μ 4 4 m 2 s ( x − x 0 ) − 21 4 + 2 s − 12393 4 a μ 1 μ 2 μ 4 4 m 2 s 2 ( x − x 0 ) − 21 4 + 2 s − 5832 a μ 1 μ 2 μ 4 4 m 2 s 3 ( x − x 0 ) − 21 4 + 2 s − 1215 2 a 3 μ 1 2 μ 4 2 m 2 x ( x − x 0 ) − 19 4 + 2 s + 324 a 3 μ 1 2 μ 4 2 m 2 s x ( x − x 0 ) − 19 4 + 2 s + 945 a 4 μ 1 μ 4 m 2 ( x − x 0 ) − 9 2 + 2 s − 810 a 3 μ 1 2 μ 4 2 m 2 ( x − x 0 ) − 15 4 + 2 s − 490 a 4 m 3 ( x − x 0 ) − 21 4 + 3 s − 2187 32 μ 1 μ 2 μ 4 4 m 3 ( x − x 0 ) − 21 4 + 3 s + 9477 8 μ 1 μ 2 μ 4 4 m 3 s ( x − x 0 ) − 21 4 + 3 s − 9477 2 μ 1 μ 2 μ 4 4 m 3 s 2 ( x − x 0 ) − 21 4 + 3 s + 4374 μ 1 μ 2 μ 4 4 m 3 s 3 ( x − x 0 ) − 21 4 + 3 s − 1215 2 a 2 μ 1 2 μ 4 2 m 3 x ( x − x 0 ) − 19 4 + 3 s + 486 a 2 μ 1 2 μ 4 2 m 3 s x ( x − x 0 ) − 19 4 + 3 s + 1260 a 3 μ 1 μ 4 m 3 ( x − x 0 ) − 9 2 + 3 s − 810 a 2 μ 1 2 μ 4 2 m 3 ( x − x 0 ) − 15 4 + 3 s − 490 a 3 m 4 ( x − x 0 ) − 21 4 + 4 s − 1215 4 a μ 1 2 μ 4 2 m 4 x ( x − x 0 ) − 19 4 + 4 s + 324 a μ 1 2 μ 4 2 m 4 s x ( x − x 0 ) − 19 4 + 4 s + 945 a 2 μ 1 μ 4 m 4 ( x − x 0 ) − 9 2 + 4 s − 405 a μ 1 2 μ 4 2 m 4 ( x − x 0 ) − 15 4 + 4 s − 294 a 2 m 5 ( x − x 0 ) − 21 4 + 5 s − 243 4 μ 1 2 μ 4 2 m 5 x ( x − x 0 ) − 19 4 + 5 s + 81 μ 1 2 μ 4 2 m 5 s x ( x − x 0 ) − 19 4 + 5 s + 378 a μ 1 μ 4 m 5 ( x − x 0 ) − 9 2 + 5 s − 81 μ 1 2 μ 4 2 m 5 ( x − x 0 ) − 15 4 + 5 s − 98 a m 6 ( x − x 0 ) − 21 4 + 6 s + 63 μ 1 μ 4 m 6 ( x − x 0 ) − 9 2 + 6 s − 14 m 7 ( x − x 0 ) − 21 4 + 7 s .

Appendix B: Computation of Ψ2(ζ, ω) for the series solution of Eq. (5.21)

Ψ 2 ( ζ , ω ) = 1 216 μ 1 μ 2 ( μ 4 ϕ ω + μ 3 ϕ ζ ) 2 × μ 1 2 Ψ 1 ( ζ , ω ) 2 + 144 μ 1 μ 2 μ 4 2 Ψ 1 ω ϕ ω + 24 μ 2 μ 4 ω ϕ ω 2 + 432 μ 1 μ 2 μ 4 2 Ψ 1 ( ζ ω ) ϕ ω ω − 52488 μ 2 2 μ 4 4 ϕ ω ω 2 − 23328 μ 2 2 μ 4 4 ϕ ω ϕ ω ω ω + 144 μ 1 μ 2 μ 3 μ 4 ϕ ω Ψ 1 ζ + 144 μ 1 μ 2 μ 3 μ 4 Ψ 1 ω ϕ ζ + 96 μ 2 μ 4 ζ ϕ ω ϕ ζ + 96 μ 2 μ 3 ω ϕ ω ϕ ζ − 23328 μ 2 2 μ 3 μ 4 3 ϕ ω ω ω ϕ ζ + 144 μ 1 μ 2 μ 3 2 Ψ 1 ζ ϕ ζ + 48 μ 2 μ 3 ζ ϕ ζ 2 + 864 μ 1 μ 2 μ 3 μ 4 Ψ 1 ( ζ ω ) ϕ ζ ω − 209952 μ 2 2 μ 3 μ 4 3 ϕ ω ω ϕ ζ ω − 209952 μ 2 2 μ 3 2 μ 4 2 ϕ ζ ω 2 − 69984 μ 2 2 μ 3 μ 4 3 ϕ ω ϕ ζ ω ω − 69984 μ 2 2 μ 3 2 μ 4 2 ϕ ζ ϕ ζ ω ω + 432 μ 1 μ 2 μ 3 2 Ψ 1 ( ζ ω ) ϕ ζ ζ − 104976 μ 2 2 μ 3 2 μ 4 2 ϕ ω ω ϕ ζ ζ − 209952 μ 2 2 μ 3 3 μ 4 ϕ ζ ω ϕ ζ ζ − 52488 μ 2 2 μ 3 4 ϕ ζ ζ 2 − 69984 μ 2 2 μ 3 2 μ 4 2 ϕ ζ ϕ ζ ζ ω − 69984 μ 2 2 μ 3 3 μ 4 ϕ ζ ϕ ζ ζ ω − 23328 μ 2 2 μ 3 3 μ 4 ϕ ω ϕ ζ ζ ζ − 23328 μ 2 2 μ 3 4 ϕ ζ ϕ ζ ζ ζ .

Appendix C: Computation of coefficients of the Right Painlevé series solution for Eq. (1.1)

U 3 ( t , x , y , z ) = 1 24 μ 1 μ 2 τ x 4 × 2 μ 1 μ 4 U 1 ( t , x , y , z ) τ z 2 + 2 μ 1 μ 3 U 1 ( t , x , y , z ) τ y 2 − 4 μ 1 2 U 1 ( t , x , y , z ) U 1 x τ x + 2 μ 1 2 U 1 ( t , x , y , z ) U 2 ( t , x , y , z ) τ x 2 + 24 μ 2 μ 4 τ z z τ x 2 + 24 μ 2 μ 3 τ y y τ x 2 + 48 μ 1 μ 2 U 2 x τ x 3 + 96 μ 2 μ 4 τ z τ x τ x z + 96 μ 2 μ 3 τ y τ x τ x y − μ 1 2 U 1 ( t , x , y , z ) 2 τ x x − 24 μ 1 μ 2 U 1 x τ x τ x x + 120 μ 1 μ 2 U 2 ( t , x , y , z ) τ x 2 τ x x − 18 μ 1 μ 2 U 1 ( t , x , y , z ) τ x x 2 + 288 μ 2 2 τ x x 3 − 16 μ 1 μ 2 U 1 ( t , x , y , z ) τ x τ x x x + 1056 μ 2 2 τ x τ x x τ x x x + 216 μ 2 2 τ x 2 τ x x x x + 2 μ 1 U 1 ( t , x , y , z ) τ x τ t + 48 μ 2 τ x τ x x τ t + 72 μ 2 τ x 2 τ t x

U 5 ( t , x , y , z ) = 1 2 τ x 2 ( μ 1 μ 2 U 1 + τ x x ) − 2 μ 4 U 3 z τ z − 2 μ 4 U 4 ( t , x , y , z ) τ z 2 − μ 4 U 2 z z − μ 4 U 3 ( t , x , y , z ) τ z z − 2 μ 3 U 3 y τ y − 2 μ 3 U 4 ( t , x , y , z ) τ y 2 − μ 3 U 2 y y − μ 3 U 3 ( t , x , y , z ) τ y y − μ 1 U 2 x 2 − 2 μ 1 U 1 x U 3 x − 2 μ 1 U 4 ( t , x , y , z ) U 1 x τ x − 2 μ 1 U 3 ( t , x , y , z ) U 2 x τ x − 2 μ 1 U 2 ( t , x , y , z ) U 3 x τ x − 2 μ 1 U 1 ( t , x , y , z ) U 4 x τ x − μ 1 U 3 ( t , x , y , z ) 2 τ x 2 − 2 μ 1 U 2 ( t , x , y , z ) U 4 ( t , x , y , z ) τ x 2 − μ 1 U 3 ( t , x , y , z ) U 1 x x − μ 1 U 2 ( t , x , y , z ) U 2 x x − μ 1 U 1 ( t , x , y , z ) U 3 x x − μ 1 U 2 ( t , x , y , z ) U 3 ( t , x , y , z ) τ x x − μ 1 U 1 ( t , x , y , z ) U 4 ( t , x , y , z ) τ x x + 24 μ 2 U 4 x τ x τ x x − 6 μ 2 U 3 x x τ x x + 18 μ 2 U 4 ( t , x , y , z ) τ x x 2 − 4 μ 2 τ x U 3 x x x − 4 μ 2 U 3 x τ x x x + 16 μ 2 U 4 ( t , x , y , z ) τ x τ x x x − μ 2 U 2 x x x x − μ 2 U 3 ( t , x , y , z ) τ x x x x − τ x U 3 t − U 3 x τ t − 2 U 4 ( t , x , y , z ) τ x τ t − U 2 t x − U 3 ( t , x , y , z ) τ t x

U 7 ( t , x , y , z ) = 1 48 μ 2 τ x 4 − 4 μ 4 U 4 z τ z − 6 μ 4 U 5 ( t , x , y , z ) τ z 2 − μ 4 U 3 z z − 2 μ 4 U 4 ( t , x , y , z ) τ z z − 4 μ 3 U 4 y τ y − 6 μ 3 U 5 ( t , x , y , z ) τ y 2 − μ 3 U 3 y y − 2 μ 3 U 4 ( t , x , y , z ) τ y y − 2 μ 1 U 2 x U 3 x − 2 μ 1 U 1 x U 4 x − 4 μ 1 U 5 ( t , x , y , z ) U 1 x τ x − 4 μ 1 U 4 ( t , x , y , z ) U 2 x τ x − 4 μ 1 U 3 ( t , x , y , z ) U 3 x τ x − 4 μ 1 U 2 ( t , x , y , z ) U 4 x τ x − 4 μ 1 U 1 ( t , x , y , z ) U 5 x τ x − 6 μ 1 U 3 ( t , x , y , z ) U 4 ( t , x , y , z ) τ x 2 − 6 μ 1 U 2 ( t , x , y , z ) U 5 ( t , x , y , z ) τ x 2 − 6 μ 1 U 1 ( t , x , y , z ) U 6 ( t , x , y , z ) τ x 2 − 48 μ 2 U 6 x τ x 3 − μ 1 U 4 ( t , x , y , z ) U 1 x x − μ 1 U 3 ( t , x , y , z ) U 2 x x − μ 1 U 2 ( t , x , y , z ) U 3 x x − μ 1 U 1 ( t , x , y , z ) U 4 x x − 24 μ 2 τ x 2 U 5 x x − μ 1 U 3 ( t , x , y , z ) 2 τ x x − 2 μ 1 U 2 ( t , x , y , z ) U 4 ( t , x , y , z ) τ x x − 2 μ 1 U 1 ( t , x , y , z ) U 5 ( t , x , y , z ) τ x x − 24 μ 2 U 5 x τ x τ x x − 24 μ 2 U 6 ( t , x , y , z ) τ x 2 τ x x − 12 μ 2 U 4 x x τ x x + 6 μ 2 U 5 ( t , x , y , z ) τ x x 2 − 8 μ 2 τ x U 4 x x x − 8 μ 2 U 4 x τ x x x − μ 2 U 3 x x x x − 2 μ 2 U 4 ( t , x , y , z ) τ x x x x − 2 τ x U 4 t − 2 U 4 x τ t − 6 U 5 ( t , x , y , z ) τ x τ t − U 3 t x − 2 U 4 ( t , x , y , z ) τ t x

U 8 ( t , x , y , z ) = 1 216 μ 2 τ x 4 × − 6 μ 4 U 5 z τ z − 12 μ 4 U 6 ( t , x , y , z ) τ z 2 − μ 4 U 4 z z − 3 μ 4 U 5 ( t , x , y , z ) τ z z − 6 μ 3 U 5 y τ y − 12 μ 3 U 6 ( t , x , y , z ) τ y 2 − μ 3 U 4 y y − 3 μ 3 U 5 ( t , x , y , z ) τ y y − μ 1 U 3 x 2 − 2 μ 1 U 2 x U 4 x − 2 μ 1 U 1 x U 5 x − 6 μ 1 U 6 ( t , x , y , z ) U 1 x τ x − 6 μ 1 U 5 ( t , x , y , z ) U 2 x τ x − 6 μ 1 U 4 ( t , x , y , z ) U 3 x τ x − 6 μ 1 U 3 ( t , x , y , z ) U 4 x τ x − 6 μ 1 U 2 ( t , x , y , z ) U 5 x τ x − 6 μ 1 U 1 ( t , x , y , z ) U 6 x τ x − 6 μ 1 U 4 ( t , x , y , z ) 2 τ x 2 − 12 μ 1 U 3 ( t , x , y , z ) U 5 ( t , x , y , z ) τ x 2 − 12 μ 1 U 2 ( t , x , y , z ) U 6 ( t , x , y , z ) τ x 2 − 12 μ 1 U 1 ( t , x , y , z ) U 7 ( t , x , y , z ) τ x 2 − 168 μ 2 U 7 x τ x 3 − μ 1 U 5 ( t , x , y , z ) U 1 x x − μ 1 U 4 ( t , x , y , z ) U 2 x x − μ 1 U 3 ( t , x , y , z ) U 3 x x − μ 1 U 2 ( t , x , y , z ) U 4 x x − μ 1 U 1 ( t , x , y , z ) U 5 x x − 60 μ 2 τ x 2 U 6 x x − 3 μ 1 U 3 ( t , x , y , z ) U 4 ( t , x , y , z ) τ x x − 3 μ 1 U 2 ( t , x , y , z ) U 5 ( t , x , y , z ) τ x x − 3 μ 1 U 1 ( t , x , y , z ) U 6 ( t , x , y , z ) τ x x − 96 μ 2 U 6 x τ x τ x x − 180 μ 2 U 7 ( t , x , y , z ) τ x 2 τ x x − 18 μ 2 U 5 x x τ x x − 12 μ 2 U 6 ( t , x , y , z ) τ x x 2 − 12 μ 2 τ x U 5 x x x − 12 μ 2 U 5 x τ x x x − 24 μ 2 U 6 ( t , x , y , z ) τ x τ x x x − μ 2 U 4 x x x x − 3 μ 2 U 5 ( t , x , y , z ) τ x x x x − 3 τ x U 5 t − 3 U 5 x τ t − 12 U 6 ( t , x , y , z ) τ x τ t − U 4 t x − 3 U 5 ( t , x , y , z ) τ t x

Appendix D: Computation of leading-order exponent and coefficient for Eq. (11.1)

0 = 5 p U 0 σ τ ( − 2 + p ) τ y 2 − 5 p 2 U 0 σ τ ( − 2 + p ) τ y 2 − 5 p U 0 σ τ ( − 1 + p ) τ y y − 30 p 2 U 0 2 σ τ ( − 3 + 2 p ) τ y τ x 2 + 30 p 3 U 0 2 σ τ ( − 3 + 2 p ) τ y τ x 2 − 30 p U 0 σ τ ( − 4 + p ) τ y τ x 3 + 55 p 2 U 0 σ τ ( − 4 + p ) τ y τ x 3 − 30 p 3 U 0 σ τ ( − 4 + p ) τ y τ x 3 + 5 p 4 U 0 σ τ ( − 4 + p ) τ y τ x 3 − 45 p 3 U 0 3 τ ( − 4 + 3 p ) τ x 4 + 45 p 4 U 0 3 τ ( − 4 + 3 p ) τ x 4 − 165 p 2 U 0 2 τ ( − 5 + 2 p ) τ x 5 + 705 2 p 3 U 0 2 τ ( − 5 + 2 p ) τ x 5 − 240 p 4 U 0 2 τ ( − 5 + 2 p ) τ x 5 + 105 2 p 5 U 0 2 τ ( − 5 + 2 p ) τ x 5 − 120 p U 0 τ ( − 6 + p ) τ x 6 + 274 p 2 U 0 τ ( − 6 + p ) τ x 6 − 225 p 3 U 0 τ ( − 6 + p ) τ x 6 + 85 p 4 U 0 τ ( − 6 + p ) τ x 6 − 15 p 5 U 0 τ ( − 6 + p ) τ x 6 + p 6 U 0 τ ( − 6 + p ) τ x 6 + 15 p 2 U 0 2 σ τ ( − 2 + 2 p ) τ x τ x y + 30 p U 0 σ τ ( − 3 + p ) τ x 2 τ x y − 45 p 2 U 0 σ τ ( − 3 + p ) τ x 2 τ x y + 15 p 3 U 0 σ τ ( − 3 + p ) τ x 2 τ x y + 15 p 2 U 0 2 σ τ ( − 2 + 2 p ) τ y τ x x + 30 p U 0 σ τ ( − 3 + p ) τ y τ x τ x x − 45 p 2 U 0 σ τ ( − 3 + p ) τ y τ x τ x x + 15 p 3 U 0 σ τ ( − 3 + p ) τ y τ x τ x x + 45 p 3 U 0 3 τ ( − 3 + 3 p ) τ x 2 τ x x + 735 2 p 2 U 0 2 τ ( − 4 + 2 p ) τ x 3 τ x x − 1215 2 p 3 U 0 2 τ ( − 4 + 2 p ) τ x 3 τ x x + 240 p 4 U 0 2 τ ( − 4 + 2 p ) τ x 3 τ x x + 360 p U 0 τ ( − 5 + p ) τ x 4 τ x x − 750 p 2 U 0 τ ( − 5 + p ) τ x 4 τ x x + 525 p 3 U 0 τ ( − 5 + p ) τ x 4 τ x x − 150 p 4 U 0 τ ( − 5 + p ) τ x 4 τ x x + 15 p 5 U 0 τ ( − 5 + p ) τ x 4 τ x x − 15 p U 0 σ τ ( − 2 + p ) τ x y τ x x + 15 p 2 U 0 σ τ ( − 2 + p ) τ x y τ x x − 315 2 p 2 U 0 2 τ ( − 3 + 2 p ) τ x τ x x 2 + 315 2 p 3 U 0 2 τ ( − 3 + 2 p ) τ x τ x x 2 − 270 p U 0 τ ( − 4 + p ) τ x 2 τ x x 2 + 495 p 2 U 0 τ ( − 4 + p ) τ x 2 τ x x 2 − 270 p 3 U 0 τ ( − 4 + p ) τ x 2 τ x x 2 + 45 p 4 U 0 τ ( − 4 + p ) τ x 2 τ x x 2 + 30 p U 0 τ ( − 3 + p ) τ x x 3 − 45 p 2 U 0 τ ( − 3 + p ) τ x x 3 + 15 p 3 U 0 τ ( − 3 + p ) τ x x 3 − 15 p U 0 σ τ ( − 2 + p ) τ x τ x x y + 15 p 2 U 0 σ τ ( − 2 + p ) τ x τ x x y − 5 p U 0 σ τ ( − 2 + p ) τ y τ x x x + 5 p 2 U 0 σ τ ( − 2 + p ) τ y τ x x x − 195 2 p 2 U 0 2 τ ( − 3 + 2 p ) τ x 2 τ x x x + 195 2 p 3 U 0 2 τ ( − 3 + 2 p ) τ x 2 τ x x x − 120 p U 0 τ ( − 4 + p ) τ x 3 τ x x x + 220 p 2 U 0 τ ( − 4 + p ) τ x 3 τ x x x − 120 p 3 U 0 τ ( − 4 + p ) τ x 3 τ x x x + 20 p 4 U 0 τ ( − 4 + p ) τ x 3 τ x x x + 75 2 p 2 U 0 2 τ ( − 2 + 2 p ) τ x x τ x x x + 120 p U 0 τ ( − 3 + p ) τ x τ x x τ x x x − 180 p 2 U 0 τ ( − 3 + p ) τ x τ x x τ x x x + 60 p 3 U 0 τ ( − 3 + p ) τ x τ x x τ x x x − 10 p U 0 τ ( − 2 + p ) τ x x x 2 + 10 p 2 U 0 τ ( − 2 + p ) τ x x x 2 + 5 p U 0 σ τ ( − 1 + p ) τ x x x y + 15 p 2 U 0 2 τ ( − 2 + 2 p ) τ x τ x x x x + 30 p U 0 τ ( − 3 + p ) τ x 2 τ x x x x − 45 p 2 U 0 τ ( − 3 + p ) τ x 2 τ x x x x + 15 p 3 U 0 τ ( − 3 + p ) τ x 2 τ x x x x − 15 p U 0 τ ( − 2 + p ) τ x x τ x x x x + 15 p 2 U 0 τ ( − 2 + p ) τ x x τ x x x x − 6 p U 0 τ ( − 2 + p ) τ x τ x x x x x + 6 p 2 U 0 τ ( − 2 + p ) τ x τ x x x x x + p U 0 τ ( − 1 + p ) τ x x x x x x − 9 p U 0 τ ( − 2 + p ) τ x τ t + 9 p 2 U 0 τ ( − 2 + p ) τ x τ t + 9 p U 0 τ ( − 1 + p ) τ t x .

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Received: 2020-04-09
Revised: 2021-07-06
Accepted: 2021-07-20
Published Online: 2021-08-09
Published in Print: 2022-12-16

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Coordinated target tracking in sensor networks by maximizing mutual information
  4. Legendre wavelet residual approach for moving boundary problem with variable thermal physical properties
  5. Computational study of intravenous magnetic drug targeting using implanted magnetizable stent
  6. Extended logistic map for encryption of digital images
  7. Valuation of the American put option as a free boundary problem through a high-order difference scheme
  8. Power minimization of gas transmission network in fully transient state using metaheuristic methods
  9. A priori error estimates for finite element approximations to transient convection-diffusion-reaction equations in fluidized beds
  10. Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation
  11. Dynamic characteristics of supersonic turbulent free jets from four types of circular nozzles
  12. New insights into singularity analysis
  13. Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations
  14. Wavelet collocation methods for solving neutral delay differential equations
  15. Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method
  16. Solving nonlinear third-order boundary value problems based-on boundary shape functions
  17. Positive radial solutions for Dirichlet problems involving the mean curvature operator in Minkowski space
  18. Effect of outlet impeller diameter on performance prediction of centrifugal pump under single-phase and cavitation flow conditions
  19. A fractional-order ship power system: chaos and its dynamical properties
  20. Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar
  21. Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations
  22. Modelling and numerical synchronization of chaotic system with fractional-order operator
https://doi.org/10.1515/ijnsns-2020-0076
Keywords for this article
closed-form solution; similarity solutions; singularity analysis; symmetry analysis

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Coordinated target tracking in sensor networks by maximizing mutual information
  4. Legendre wavelet residual approach for moving boundary problem with variable thermal physical properties
  5. Computational study of intravenous magnetic drug targeting using implanted magnetizable stent
  6. Extended logistic map for encryption of digital images
  7. Valuation of the American put option as a free boundary problem through a high-order difference scheme
  8. Power minimization of gas transmission network in fully transient state using metaheuristic methods
  9. A priori error estimates for finite element approximations to transient convection-diffusion-reaction equations in fluidized beds
  10. Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation
  11. Dynamic characteristics of supersonic turbulent free jets from four types of circular nozzles
  12. New insights into singularity analysis
  13. Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations
  14. Wavelet collocation methods for solving neutral delay differential equations
  15. Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method
  16. Solving nonlinear third-order boundary value problems based-on boundary shape functions
  17. Positive radial solutions for Dirichlet problems involving the mean curvature operator in Minkowski space
  18. Effect of outlet impeller diameter on performance prediction of centrifugal pump under single-phase and cavitation flow conditions
  19. A fractional-order ship power system: chaos and its dynamical properties
  20. Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar
  21. Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations
  22. Modelling and numerical synchronization of chaotic system with fractional-order operator
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