Almuneef Hagag 2023b - Revision history

Rimni: /* References */

2024-01-10T12:43:51Z

‎References

Ahmedshehata at 13:14, 9 January 2024

2024-01-09T13:14:14Z

Ahmedshehata at 09:53, 9 January 2024

2024-01-09T09:53:03Z

Rimni at 15:21, 8 January 2024

2024-01-08T15:21:23Z

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Rimni: /* 5.4 Application 4 */

2024-01-08T13:15:46Z

‎5.4 Application 4

Rimni at 13:14, 8 January 2024

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Rimni at 13:12, 8 January 2024

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Rimni at 12:54, 8 January 2024

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Rimni at 13:43, 5 January 2024

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Rimni at 13:29, 5 January 2024

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 [5] Manjula M., Kaliraj K.,  Botmart T., Nisar K.S., Ravichandran C.Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses. AIMS Mathematics, 8:4645-4665, 2023.
-[6]  Alqahtani Z.,   Hagag A.E. A fractional numerical study on a plant disease model with replanting and preventive treatment. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 39(3), 27, 1-21, 2023.
+[6]  Alqahtani Z.,   Hagag A.E. A fractional numerical study on a plant disease model with replanting and preventive treatment. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 39(3), 27, 2023.
 [7]  Panda S.,  Vijayakumar V.,  Nisar K.S. Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings. Boundary Value Problems, 2023:91, 2023.
@@ Line 1,190: / Line 1,190: @@
 [9]  Cattani C. Haar wavelet-based technique for sharp jumps classification. Math. Comput. Model., 39(2-3):255-278, 2004.
-[10]  Alqahtani Z.,  Hagag A.E. A new semi-analytical solution of compound KdV-Burgers equation of fractional order. Revista internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería , 39(4), 38:1-16, 2023.
+[10]  Alqahtani Z.,  Hagag A.E. A new semi-analytical solution of compound KdV-Burgers equation of fractional order. Revista internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería , 39(4), 38, 2023.
 [11]  Khan A. et al. Exact Controllability of hilfer fractional differential system with non-instantaneous impluleses and state dependent delay, Qualitative Theory of Dynamical Systems, 22:1-19, 62, 2023.
@@ Line 1,208: / Line 1,208: @@
 [18]  Shah R. et al. Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay. Mathematics, 7(6):532, 2019.
-[19]  Almuneef A. and Hagag A.E., Approximate solution of the fractional differential equation via the natural decomposition method. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(4):1-15, 2023.
+[19]  Almuneef A. and Hagag A.E., Approximate solution of the fractional differential equation via the natural decomposition method. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(4), 43, 2023.
 [20] Elbadri M., et al. A new solution of time-fractional coupled KdV equation by using natural decomposition method.  Abstract and Applied Analysis, 2020:3950816, 2020.

@@ Line 1,208: / Line 1,208: @@
 [18]  Shah R. et al. Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay. Mathematics, 7(6):532, 2019.
-[19]  Alqahtani Z. and Hagag A.E., A new semi-analytical solution of compound KdV-Burgers equation of fractional order. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(4):1-16, 2023.
+[19]  Almuneef A. and Hagag A.E., Approximate solution of the fractional differential equation via the natural decomposition method. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(4):1-15, 2023.
 [20] Elbadri M., et al. A new solution of time-fractional coupled KdV equation by using natural decomposition method.  Abstract and Applied Analysis, 2020:3950816, 2020.

@@ Line 1,208: / Line 1,208: @@
 [18]  Shah R. et al. Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay. Mathematics, 7(6):532, 2019.
-[19]  Rawashdeh M. The fractional natural decomposition method: theories and applications. Mathematical Methods in the Applied Sciences 40 (2017) 2362-2376.
+[19]  Alqahtani Z. and Hagag A.E., A new semi-analytical solution of compound KdV-Burgers equation of fractional order. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(4):1-16, 2023.
 [20] Elbadri M., et al. A new solution of time-fractional coupled KdV equation by using natural decomposition method.  Abstract and Applied Analysis, 2020:3950816, 2020.

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 |+
 |-
-|style="text-align: center;padding:10px;width: 49%;"| [[File:4ex.jpg|222x222px]]
+|style="text-align: center;padding:10px;width: 49%;"| [[File:4ex.jpg|242px]]
-|style="text-align: center;padding:10px;width: 59%;"| [[File:4app.jpg|216x216px]]
+|style="text-align: center;padding:10px;width: 59%;"| [[File:4app.jpg|236px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
 |-
-|style="text-align: center;padding:10px;width: 49%;"| [[File:4exapp.jpg|204x204px]]
+|style="text-align: center;padding:10px;width: 49%;"| [[File:4exapp.jpg|224px]]
-|style="text-align: center;padding:10px;width: 59%;"| [[File:4ex+app.jpg|198x198px]]
+|style="text-align: center;padding:10px;width: 59%;"| [[File:4ex+app.jpg|218px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
 |-
-| style="text-align: center;padding:10px;width: 49%;"|[[File:4alphaX.jpg|228x228px]]
+| style="text-align: center;padding:10px;width: 49%;"|[[File:4alphaX.jpg|248px]]
-|  style="text-align: center;padding:10px;width: 50%;"|[[File:4alphaT.jpg|222x222px]]
+|  style="text-align: center;padding:10px;width: 50%;"|[[File:4alphaT.jpg|242px]]
 |-
 |  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>

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 |-
-|style="text-align: center;padding:10px;width: 49%;"| [[File:3ex.jpg|222x222px]]
+|style="text-align: center;padding:10px;width: 49%;"| [[File:3ex.jpg|242x242px]]
-|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image14.jpeg|216px]]
+|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image14.jpeg|236px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
 |-
-|style="text-align: center;padding:10px;width: 49%;"| [[File:3exapp.jpg|204x204px]]
+|style="text-align: center;padding:10px;width: 49%;"| [[File:3exapp.jpg|224px]]
-|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image16.jpeg|198px]]
+|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image16.jpeg|218px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
 |-
-| style="text-align: center;padding:10px;width: 49%;"|[[File:3alphaX.jpg|228x228px]]
+| style="text-align: center;padding:10px;width: 49%;"|[[File:3alphaX.jpg|248px]]
-|  style="text-align: center;padding:10px;width: 50%;"|[[File:3alphaT.jpg|222x222px]]
+|  style="text-align: center;padding:10px;width: 50%;"|[[File:3alphaT.jpg|242px]]
 |-
 |  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>

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 |  style="text-align: center;vertical-align: top;"|-4.000000000
 |}
 <div id='img-3'></div>

@@ Line 83: / Line 83: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>{J}_{a}^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{1}{\Gamma \left( \zeta +1\right) }\int_{a}^{\theta }\omega \left( \theta \right) {\left( d\theta \right) }^{\zeta }=\frac{1}{\Gamma \left( \zeta \right) }\int_{a}^{\theta }{\left( \theta -\upsilon \right) }^{\zeta -1}\omega \left( \upsilon \right) d\upsilon ,&\theta ,\zeta >0,\\\omega \left( \theta \right) ,&\zeta =0.\end{matrix}\right.</math>
+| <math>{J}_{a}^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{1}{\Gamma \left( \zeta +1\right) }\displaystyle\int_{a}^{\theta }\omega \left( \theta \right) {\left( d\theta \right) }^{\zeta }=\frac{1}{\Gamma \left( \zeta \right) }\displaystyle\int_{a}^{\theta }{\left( \theta -\upsilon \right) }^{\zeta -1}\omega \left( \upsilon \right) d\upsilon ,&\theta ,\zeta >0,\\\omega \left( \theta \right) ,&\zeta =0.\end{matrix}\right.</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(5)
@@ Line 96: / Line 96: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>{D}_{\ast }^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{{d}^{k}}{d{\theta }^{k}}\omega \left( \theta \right) ,&\zeta =k\in N,\\\frac{1}{\Gamma \left( k-\zeta \right) }\int_{0}^{\theta }{\left( \theta -\upsilon \right) }^{k-\zeta -1}{\omega }^{\left( k\right) }\left( \upsilon \right) d\upsilon ,&k-1<\zeta \leq k\in N.\end{matrix}\right.</math>
+| <math>{D}_{\ast }^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{{d}^{k}}{d{\theta }^{k}}\omega \left( \theta \right) ,&\zeta =k\displaystyle\in N,\\\frac{1}{\Gamma \left( k-\zeta \right) }\displaystyle\int_{0}^{\theta }{\left( \theta -\upsilon \right) }^{k-\zeta -1}{\omega }^{\left( k\right) }\left( \upsilon \right) d\upsilon ,&k-1<\zeta \leq k\in N.\end{matrix}\right.</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(6)

@@ Line 1,169: / Line 1,169: @@
 ==References==
-[1] R. Hilfer, Applications of Fractional Calculus in Physics, Academic press, Orlando, 1999.
+[1]  Hilfer R. Applications of fractional calculus in physics. Academic press, Orlando, 1999.
-[2] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
+[2] Kilbas  A.,  Srivastava H.,  Trujillo J. Theory and applications of fractional differential equations. Elsevier,  204:1-523, 2006.
-[3] I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999).
+[3]  Podlubny I. Fractional differential equations. Academic Press, New York, 1999.
-[4] R. Amin, et al, Solution of Variable-Order Nonlinear Fractional Differential Equations Using Haar Wavelet Collocation Technique, Fractals 31 (2023) 2340022.
+[4]  Amin R., Hafsa,  Hadi F., Altanji M., Nisar K.S., Sumelka W. Solution of variable-order nonlinear fractional differential equations using haar wavelet collocation technique. Fractals, 31(2):2340022, 2023.
-[5] M. Manjula, et al, Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses, AIMS Mathematics 8 (2023) 4645-4665.
+[5] Manjula M., Kaliraj K.,  Botmart T., Nisar K.S., Ravichandran C.Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses. AIMS Mathematics, 8:4645-4665, 2023.
-[6] Z. Alqahtani and A. E. Hagag, A fractional numerical study on a plant disease model with replanting and preventive treatment. Métodos numéricos para cálculo y diseño en ingeniería: Revista internacional, 39(3) (2023) 1-21.
+[6]  Alqahtani Z.,   Hagag A.E. A fractional numerical study on a plant disease model with replanting and preventive treatment. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 39(3), 27, 1-21, 2023.
 [7] S. Panda, V. Vijayakumar, and K. S. Nisar, Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings, Boundary Value Problems 2023 (2023) 91.

@@ Line 1,156: / Line 1,156: @@
 |}
-==6 Conclusion==
+==6. Conclusion==
 Studying and exploring nonlinear physical models using new techniques always help us advance in science and technology. In the current framework, we used NDM to evaluate the BBMPB equation with fractional order. Banach’s fixed-point theory is used to investigate the anticipated issue’s uniqueness theorem and convergence analysis. The anticipated method’s dependability and applicability are demonstrated by presenting four cases. The behaviors for the obtained findings are provided in 2D, 3D graphs, and tables for featured fractional order. These graphs aid to conclude the stimulating behaviors of the analogical models. Furthermore, while solving nonlinear issues, NDM does not require any conversion, perturbation, or consideration of extra polynomials or parameters. The examination of these kinds of occurrences can provide new ideas for investigating more real-world events. It can also generate ideas for employing an accurate method to evaluate nonlinear models related to science and technology. This work elucidates the proposed model, which is notably dependent on time instant and its history and can be convincingly illustrated utilizing fractional notions.
@@ Line 1,164: / Line 1,164: @@
 '''Data Availability:''' No data were used to support this study.
-'''Acknowledgments: '''This research project was funded by the Deanship of Scientific Research, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant No (43- PRFA-P-43).
+==Acknowledgments==
 ==References==