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Erratum to “Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications”

  • Pinki Kumari , R. K. Gupta , Sachin Kumar EMAIL logo and Maysaa Mohamed Al Qurashi
Published/Copyright: December 29, 2020
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Open Physics
From the journal Open Physics Volume 18 Issue 1

Abstract

This erratum corrects the typing mistakes of the article “Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications,” published in Open Physics 2020;18(1):164–9, https://doi.org/10.1515/phys-2020-0127.

Keywords: systems of partial differential equations; Lie symmetry; conformable derivative; conservation laws

1 The dispersive long-wave system

In ref. [1], the classical dispersive long-wave system is given by

(1) ∂ u ∂ t = ( u 2 − u x + 2 v ) x , ∂ v ∂ t = ( 2 u v + v x ) x ,

which is used to describe evolution of the horizontal velocity portion of water waves. We present here the correct form of (1) with conformable operator as

(2) ∂ α u ∂ t α = 2 u ∂ α u ∂ x α − ∂ 2 α u ∂ x 2 α + 2 ∂ α v ∂ x α , ∂ α v ∂ t α = 2 u ∂ α v ∂ x α + 2 v ∂ α u ∂ x α + ∂ 2 α v ∂ x 2 α .

We present the symmetries of the correct version of conformable dispersive long-wave system described in (2) as

(3) ξ 1 = c 1 t α + c 2 t 1 − α , ξ 2 = c 1 2 α x + c 4 x 1 − α + c 3 α x 1 − α t α , η 1 = − c 1 2 u − c 3 2 , η 2 = − c 1 v .

Thus, the Lie algebra of (2) is spanned by the following four generators:

(4) X 1 = t α ∂ ∂ t + x 2 α ∂ ∂ x − 1 2 u ∂ ∂ u − v ∂ ∂ v , X 2 = t 1 − α ∂ ∂ t , X 3 = x 1 − α t α α ∂ ∂ x − 1 2 ∂ ∂ u , X 4 = x 1 − α ∂ ∂ x .

Below, we will present the correct form of conserved vectors.

For the symmetry X 1 = t α ∂ ∂ t + x 2 α ∂ ∂ x − 1 2 u ∂ ∂ u − v ∂ ∂ v , we obtain

(5) T 1 t = c 1 x α − 1 − 1 2 u − x 2 α u x − t α u t + c 2 x α − 1 − v − x 2 α v x − t α v t , T 1 x = 2 − 1 2 u − x 2 α u x − t α u t ( c 1 u − c 2 v ) t 1 − α − c 1 1 2 u x + x 2 α u x x + 1 2 α u x + 1 α u t x x 1 − α t α − 1 − − v − x 2 α v x − t α v t ( 2 c 1 + 2 c 2 u + c 2 x − α + c 2 ( 1 − α ) x − α ) t α − 1 + c 2 v x + x 2 α v x x + 1 2 α v x + 1 α v t x x 1 − α t α − 1 .

For the symmetry X 2 = t 1 − α ∂ ∂ t , we obtain

(6) T 1 t = − c 1 x α − 1 t 1 − α u t − c 2 x α − 1 t 1 − α v t , T 1 x = − t 1 − α u t ( c 1 u − c 2 v ) t 1 − α − c 1 t 1 − α u t x x 1 − α t α − 1 + t 1 − α v t ( 2 c 1 + 2 c 2 u + c 2 x − α + c 2 ( 1 − α ) x − α ) t α − 1 + c 2 v t x x 1 − α .

For the symmetry X 3 = x 1 − α t α α ∂ ∂ x − 1 2 ∂ ∂ u , we obtain

(7) T 1 t = c 1 x α − 1 − 1 2 − x 1 − α t α α u x − c 2 t α α v x , T 1 x = − 1 2 − x 1 − α t α α u x ( c 1 u − c 2 v ) t 1 − α − c 1 ( 1 − α ) α u x + 1 α x u x x x 1 − 2 α t 2 α − 1 + x 1 − α t α α v x ( 2 c 1 + 2 c 2 u + c 2 x − α + c 2 ( 1 − α ) x − α ) t α − 1 + c 2 ( 1 − α ) α v x + 1 α x v x x x 1 − 2 α t 2 α − 1 .

For the symmetry X 4 = x 1 − α ∂ ∂ x , we obtain

(8) T 1 t = − c 1 u x − c 2 v x , T 1 x = − x 1 − α u x ( c 1 u − c 2 v ) t 1 − α − c 1 ( ( 1 − α ) u x + x u x x ) x 1 − 2 α t α − 1 + x 1 − α v x ( 2 c 1 + 2 c 2 u + c 2 x − α + c 2 ( 1 − α ) x − α ) t α − 1 + c 2 ( ( 1 − α ) v x + x v x x ) x 1 − 2 α t α − 1 .

2 The Whitham–Broer–Kaup system

The conformable Whitham–Broer–Kaup Wilson system used in ref. [1] is given by

(9) ∂ α u ∂ t α + u ∂ α u ∂ x α + μ ∂ 2 α u ∂ x 2 α + ∂ α v ∂ x α = 0 , ∂ α v ∂ t α + u ∂ α v ∂ x α + v ∂ α u ∂ x α − μ ∂ 2 α v ∂ x 2 α + β ∂ 3 α v ∂ x 3 α = 0 .

We present the following symmetries for system (9)

(10) ξ 1 = c 1 t α + c 2 t 1 − α , ξ 2 = c 1 2 α x + c 4 x 1 − α + c 3 α x 1 − α t α , η 1 = − c 1 2 u + c 3 , η 2 = − c 1 v .

Thus, the Lie algebra of (9) is spanned by the following four generators

(11) X 1 = t α ∂ ∂ t + x 2 α ∂ ∂ x − 1 2 u ∂ ∂ u − v ∂ ∂ v , X 2 = t 1 − α ∂ ∂ t , X 3 = x 1 − α t α α ∂ ∂ x + ∂ ∂ u , X 4 = x 1 − α ∂ ∂ x .

In the next step, we present the correct version of the conserved vectors for system (9).

For the symmetry X 1 = t α ∂ ∂ t + x 2 α ∂ ∂ x − 1 2 u ∂ ∂ u − v ∂ ∂ v , we obtain

(12) T 1 t = c 1 x α − 1 − 1 2 u − x 2 α u x − t α u t + c 2 x α − 1 − v − x 2 α v x − t α v t , T 1 x = − 1 2 u − x 2 α u x − t α u t ( c 1 u + c 1 μ c 2 x − α + c 2 v + c 1 β (1 − α )(1 − 2 α ) x − 2 α − c 1 μ (1 − α ) x − α − 3 c 2 β (1 − α ) 2 x − α + c 2 β (2 − 2 α )(1 − α ) x − α ) t α − 1 − 1 2 u x + x 2 α u x x + 1 2 α u x + 1 α u t x × ( c 1 μ x 1 − α + 3 β (1 − α ) c 2 x 1 − 2 α − c 2 β x 2 − 2 α ) t α − 1 − 1 2 u x x + 1 2 α u x x + x 2 α u x x x + 1 2 α u x x + 1 α u t x x × c 2 x 2 − 2 α t α − 1 + − v − x 2 α v x − t α v t × ( c 1 + c 2 u − c 2 μ (1 − α ) x 1 − α + c 2 μ (1 − α ) x − α ) t α − 1 − v x + x 2 α v x x + 1 2 α v x + 1 α v t x c 2 μ x 1 − α t α − 1 .

For the symmetry X 2 = t 1 − α ∂ ∂ t , we obtain

(13) T 1 t = − c 1 x α − 1 t 1 − α u t − c 2 x α − 1 t 1 − α v t , T 1 x = − u t ( c 1 u + c 1 μ c 2 x − α + c 2 v + c 1 β ( 1 − α ) × ( 1 − 2 α ) x − 2 α − c 1 μ ( 1 − α ) x − α − 3 c 2 β ( 1 − α ) 2 x − α + c 2 β ( 2 − 2 α ) ( 1 − α ) ) x − α − u t x ( c 1 μ x 1 − α + 3 β ( 1 − α ) c 2 x 1 − 2 α − c 2 β x 2 − 2 α ) − u t x x c 2 x 2 − 2 α − v t ( c 1 + c 2 u − c 2 μ ( 1 − α ) x 1 − α + c 2 μ ( 1 − α ) x − α ) − c 2 μ x 1 − α v t x .

For the symmetry X 3 = x 1 − α t α α ∂ ∂ x + ∂ ∂ u , we obtain

(14) T 1 t = c 1 x α − 1 − 1 − x 1 − α t α α u x − c 2 t α α v x , T 1 x = − 1 − x 1 − α t α α u x ( c 1 u + c 1 μ c 2 x − α + c 2 v + c 1 β (1 − α )(1 − 2 α ) x − 2 α − c 1 μ (1 − α ) x − α − 3 c 2 β (1 − α ) 2 x − α + c 2 β (2 − 2 α )(1 − α ) ) x − α t α − 1 − (1 − α ) α u x + 1 α x u x x ( c 1 μ x 1 − α + 3 β (1 − α ) c 2 x 1 − 2 α − c 2 β x 2 − 2 α ) t 2 α − 2 − (1 − α ) α u x x + 1 α x u x x x + 1 α u x x × c 2 x 2 − 2 α t α − 1 + x 1 − α t α α v x (2 c 1 + 2 c 2 u + c 2 x − α + c 2 (1 − α ) x − α ) t α − 1 + c 2 μ (1 − α ) α v x + 1 α x v x x x 1 − 2 α t 2 α − 1 .

For the symmetry X 4 = x 1 − α ∂ ∂ x , we obtain

(15) T 1 t = − c 1 u x − c 2 v x , T 1 x = − x 1 − α u x ( c 1 u + c 1 μ c 2 x − α + c 2 v + c 1 β ( 1 − α ) × ( 1 − 2 α ) x − 2 α − c 1 μ ( 1 − α ) x − α − 3 c 2 β ( 1 − α ) 2 × x − α + c 2 β ( 2 − 2 α ) ( 1 − α ) x − α ) t α − 1 − ( ( 1 − α ) x − α u x + x 1 − α u x x ) × ( c 1 μ x 1 − α + 3 β ( 1 − α ) c 2 x 1 − 2 α − c 2 β x 2 − 2 α ) t α − 1 − ( 2 ( 1 − α ) x − α u x x − α ( 1 − α ) x − α − 1 u x + x 1 − α u x x x ) c 2 x 2 − 2 α t α − 1 − x 1 − α v x ( c 1 + c 2 u − c 2 μ ( 1 − α ) x 1 − α + c 2 μ ( 1 − α ) x − α ) t α − 1 − ( ( 1 − α ) x − α v x + x 1 − α v x x ) c 2 μ x 1 − α t α − 1 .

Reference

[1] Al Qurashi MM. Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications. Open Phys. 2020;18(1):164–9.10.1515/phys-2020-0127Search in Google Scholar
Received: 2020-09-27
Accepted: 2020-11-27
Published Online: 2020-12-29

© 2020 Pinki Kumari et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2020-0219
Keywords for this article
systems of partial differential equations; Lie symmetry; conformable derivative; conservation laws
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  107. Joint use of eddy current imaging and fuzzy similarities to assess the integrity of steel plates
  108. Uncertainty quantification in the design of wireless power transfer systems
  109. Influence of unequal stator tooth width on the performance of outer-rotor permanent magnet machines
  110. New elements within finite element modeling of magnetostriction phenomenon in BLDC motor
  111. Evaluation of localized heat transfer coefficient for induction heating apparatus by thermal fluid analysis based on the HSMAC method
  112. Experimental set up for magnetomechanical measurements with a closed flux path sample
  113. Influence of the earth connections of the PWM drive on the voltage constraints endured by the motor insulation
  114. High temperature machine: Characterization of materials for the electrical insulation
  115. Architecture choices for high-temperature synchronous machines
  116. Analytical study of air-gap surface force – application to electrical machines
  117. High-power density induction machines with increased windings temperature
  118. Influence of modern magnetic and insulation materials on dimensions and losses of large induction machines
  119. New emotional model environment for navigation in a virtual reality
  120. Performance comparison of axial-flux switched reluctance machines with non-oriented and grain-oriented electrical steel rotors
  121. Erratum
  122. Erratum to “Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications”
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