Almuneef Hagag 2023a - Revision history

Rimni at 08:44, 22 November 2023

2023-11-22T08:44:29Z

Ahmedshehata at 08:44, 22 November 2023

2023-11-22T08:44:16Z

Ahmedshehata at 08:28, 22 November 2023

2023-11-22T08:28:20Z

Rimni at 13:35, 17 November 2023

2023-11-17T13:35:39Z

Rimni at 13:10, 17 November 2023

2023-11-17T13:10:46Z

Rimni: /* References */

2023-11-17T11:50:43Z

‎References

Rimni at 15:22, 16 November 2023

2023-11-16T15:22:24Z

Rimni at 15:04, 15 November 2023

2023-11-15T15:04:15Z

Rimni at 14:55, 15 November 2023

2023-11-15T14:55:42Z

Rimni at 14:54, 15 November 2023

2023-11-15T14:54:31Z

@@ Line 30: / Line 30: @@
 |}
-where <math display="inline">\epsilon</math>  is the time fractional derivative. There are a few publications available about this equation. In 2007,  Ugurlu and [28] considered the developed tanh function technique to discover some exact solutions to the CRWP equation. In 2013, Bulut [27]  solved the CRWP problem using hyperbolic functions with the Kudryashov technique. Many novel solutions have been found by Odabasi and Misirli [29] by using the modified trial equation technique. Guner et al. [30] used two efficient techniques, namely the (G’/G,1/G)-expansion method and the (G’/G)-expansion method. In 2021, Bashar et al investigated novel exact solutions of the CRWP equation utilizing an expanded version of the Exp (−φ(ζ))-expansion technique in the sense of conformable fractional derivative [31]. By using the fractional natural decomposition technique (FNDM), this article finds approximate analytical solutions to the (CRWP) equation and compares them to the exact solutions.
+where <math display="inline">\epsilon</math>  is the time fractional derivative. There are a few publications available about this equation. In 2007,  Ugurlu and [28] considered the developed tanh function technique to discover some exact solutions to the CRWP equation. In 2013, Bulut [27]  solved the CRWP problem using hyperbolic functions with the Kudryashov technique. Many novel solutions have been found by Odabasi and Misirli [29] by using the modified trial equation technique. Guner et al. [30] used two efficient techniques, namely the (G’/G,1/G)-expansion method and the (G’/G)-expansion method. In 2021, Bashar et al. investigated novel exact solutions of the CRWP equation utilizing an expanded version of the Exp (−φ(ζ))-expansion technique in the sense of conformable fractional derivative [31]. By using the fractional natural decomposition technique (FNDM), this article finds approximate analytical solutions to the (CRWP) equation and compares them to the exact solutions.
 The remainder of this article is arranged according to the following: In Section 2, we explain the basic concept of fractional calculus and certain preliminaries that pertain to our work. A fractional natural decomposition method (FNDM) is discussed in Section 3. The convergence analysis of NDM is discussed in Section 4. In Section 5, we will solve the (CRWP) problem using the technique described above. To conclude, there is an explanation of the major findings and a comparison of them to their exact solutions.

@@ Line 1,088: / Line 1,088: @@
 [5] West B.J., Bologna M., Grigolini P., West B.J., Bologna M., Grigolini P. Fractional laplace transforms. Physics of Fractal Operators, pp. 157-183, 2003.
-[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.
+[6] Alqahtani, Z. and Hagag, A.E. 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):1-21, 2023.
-[7] Jesus I.S., ¡ Tenreiro Machado J.A. Fractional control of heat diffusion systems. Nonlinear Dynamics, 54:263-282, 2008.
+[7] 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.
 [8] Feng Z. The first-integral method to study the Burgers–Korteweg–de Vries equation. Journal of Physics A: Mathematical and General, 35(2):343, 2002.

@@ Line 30: / Line 30: @@
 |}
-where <math display="inline">\epsilon</math>  is the time fractional derivative. There are a few publications available about this equation. In 2007,  Ugurlu and [28] considered the developed tanh function technique to discover some exact solutions to the CRWP equation. In 2013, Bulut [27]  solved the CRWP problem using hyperbolic functions with the Kudryashov technique. Many novel solutions have been found by Odabasi and Misirli [29] by using the modified trial equation technique. Guner et al. [30] used two efficient techniques, namely the (G’/G,1/G)-expansion method and the (G’/G)-expansion method. In 2019, Dipankar Kumar and Samir Chandra Ray [31] investigated novel exact solutions of the CRWP equation utilizing an expanded version of the Exp (−φ(ζ))-expansion technique in the sense of conformable fractional derivative. By using the fractional natural decomposition technique (FNDM), this article finds approximate analytical solutions to the (CRWP) equation and compares them to the exact solutions.
+where <math display="inline">\epsilon</math>  is the time fractional derivative. There are a few publications available about this equation. In 2007,  Ugurlu and [28] considered the developed tanh function technique to discover some exact solutions to the CRWP equation. In 2013, Bulut [27]  solved the CRWP problem using hyperbolic functions with the Kudryashov technique. Many novel solutions have been found by Odabasi and Misirli [29] by using the modified trial equation technique. Guner et al. [30] used two efficient techniques, namely the (G’/G,1/G)-expansion method and the (G’/G)-expansion method. In 2021, Bashar et al investigated novel exact solutions of the CRWP equation utilizing an expanded version of the Exp (−φ(ζ))-expansion technique in the sense of conformable fractional derivative [31]. By using the fractional natural decomposition technique (FNDM), this article finds approximate analytical solutions to the (CRWP) equation and compares them to the exact solutions.
 The remainder of this article is arranged according to the following: In Section 2, we explain the basic concept of fractional calculus and certain preliminaries that pertain to our work. A fractional natural decomposition method (FNDM) is discussed in Section 3. The convergence analysis of NDM is discussed in Section 4. In Section 5, we will solve the (CRWP) problem using the technique described above. To conclude, there is an explanation of the major findings and a comparison of them to their exact solutions.

@@ Line 123: / Line 123: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>\begin{matrix}{D}_{\tau }^{\epsilon }\varphi \left( x,\tau \right) +R\varphi \left( x,\tau \right) +F\varphi \left( x,\tau \right) &=&\hslash \left( x,\tau \right), n-1<\epsilon \leq n,\end{matrix}</math>
+| <math>{D}_{\tau }^{\epsilon }\varphi \left( x,\tau \right) +R\varphi \left( x,\tau \right) +F\varphi \left( x,\tau \right) =\hslash \left( x,\tau \right), n-1<\epsilon \leq n,</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
@@ Line 135: / Line 135: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>\begin{matrix}\varphi \left( x,0\right) &=&v\left( x\right) ,\end{matrix}</math>
+| <math>\varphi \left( x,0\right) =v\left( x\right) ,</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(8)

@@ Line 1,078: / Line 1,078: @@
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Miller K.S.,  Ross B. An introduction to the fractional calculus and fractional differential equations. 1993
+[1] Miller K.S.,  Ross B. An introduction to the fractional calculus and fractional differential equations.  Wiley-Interscience, 1st edition, pp. 384, 1993.
-[2] Kilbas A.A., Srivastava H.M.,  Trujillo J.J. Theory and applications of fractional differential equations. (Vol. 204). elsevier, 2006.
+[2] Kilbas A.A., Srivastava H.M.,  Trujillo J.J. Theory and applications of fractional differential equations. Elsevier, Vol. 204, pp. 523, 2006.
-[3] Kilbas A.A., Marichev O.I.,  Samko S.G. Fractional integrals and derivatives (theory and applications).  1993
+[3] Samko S.G., Kilbas A.A., Marichev O.I.   Fractional integrals and derivatives: Theory and applications.  Gordon and Breach Science Publishers,  pp. 976, 1993.
 [4] Herzallah M.A., Muslih S.I., Baleanu D., Rabei E.M. Hamilton–Jacobi and fractional like action with time scaling. Nonlinear Dynamics, 66:549-555, 2011.
@@ Line 1,136: / Line 1,136: @@
 [29] Odabasi M., Misirli E. On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences, 41(3):904-911, 2018.
-[30] Guner, O., Bekir, A. and Ünsal, Ö., 2016. Two reliable methods for solving the time fractional Clannish Random Walker's Parabolic equation. Optik, 127(20), pp.9571-9577.
+[30] Guner O., Bekir A., Ünsal Ö. Two reliable methods for solving the time fractional Clannish Random Walker's Parabolic equation. Optik, 127(20):9571-9577, 2016.
-[31] Bashar, M.H., Tahseen, T. and SHAHEN, N.H., 2021. Application of the Advanced exp (-φ (ξ))-expansion method to the Nonlinear Conformable Time-Fractional Partial Differential Equations. Turkish Journal of Mathematics and Computer Science, 13(1), pp.68-80.
+[31] Bashar M.H., Tahseen T.,  Shahen N.H. Application of the advanced exp (-φ (ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations. Turkish Journal of Mathematics and Computer Science, 13(1):68-80, 2021.
-[32] Podlubny, I., 1998. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.
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-[33] Mainardi, F., 1994. On the initial value problem for the fractional diffusion-wave equation. Waves and Stability in Continuous Media, World Scientific, Singapore, 1994, pp.246-251.
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-[34] Loonker, D. and Banerji, P.K., 2013. Solution of fractional ordinary differential equations by natural transform. Int. J. Math. Eng. Sci, 12(2), pp.1-7.
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-[35] Singh, P. and Sharma, D., 2018. Convergence and error analysis of series solution of nonlinear partial differential equation. Nonlinear Engineering, 7(4), pp.303-308.
+[36] Kreyszig E. Introductory functional analysis with applications.  John Wiley & Sons, Vol. 17, pp. 704, 1991.
-−
-[36] Kreyszig, E., 1991. Introductory functional analysis with applications (Vol. 17). John Wiley & Sons.

@@ Line 1,108: / Line 1,108: @@
 [15] Jawad A.J.A.M., Petković M.D., Biswas A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation, 217(2):869-877, 2010.
-[16] Kilic B. Improved šG 0/GŽ-expansion method for the time-fractional biological population model and Cahn–Hilliard equation. Journal of Computational and Nonlinear Dynamics, 10, pp.051016-1, 2015.
+[16] Baleanu D., Uǧurlu Y., İnç M., Kilic B. Improved (G'/G)-expansion method for the time-fractional biological population model and Cahn–Hilliard equation. Journal of Computational and Nonlinear Dynamics, 10:051016-1, 2015.
-[17] Kumar, D., Hosseini, K. and Samadani, F., 2017. The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, pp.439-446.
+[17] Kumar D., Hosseini K.,  Samadani F. The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149:439-446, 2017.
-[18] Seadawy, A.R., Kumar, D. and Chakrabarty, A.K., 2018. Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method. The European Physical Journal Plus, 133(5), p.182.
+[18] Seadawy A.R., Kumar D.,  Chakrabarty A.K. Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method. The European Physical Journal Plus, 133(5):182, 2018.
-[19] Kumar, D. and Kaplan, M., 2018. New analytical solutions of (2+ 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques. Chinese journal of physics, 56(5), pp.2173-2185.
+[19] Kumar D., Kaplan M. New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques. Chinese Journal of Physics, 56(5):2173-2185, 2018.
-[20] Chen, H. and Zhang, H., 2004. New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation. Chaos, Solitons & Fractals, 19(1), pp.71-76.
+[20] Chen H.,  Zhang H. New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation. Chaos, Solitons & Fractals, 19(1):71-76, 2004.
-[21] Adomian, G., 1984. A new approach to nonlinear partial differential equations. Journal of Mathematical Analysis and Applications, 102(2), pp.420-434.
+[21] Adomian G. A new approach to nonlinear partial differential equations. Journal of Mathematical Analysis and Applications, 102(2):420-434, 1984.
-[22] Sakar, M.G., Erdogan, F. and Yıldırım, A., 2012. Variational iteration method for the time-fractional Fornberg–Whitham equation. Computers & Mathematics with Applications, 63(9), pp.1382-1388.
+[22] Sakar M.G., Erdogan F.,  Yıldırım A. Variational iteration method for the time-fractional Fornberg–Whitham equation. Computers & Mathematics with Applications, 63(9):1382-1388, 2012.
 [23] Rawashdeh M.S., Al-Jammal H. New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM. Advances in Difference Equations, 2016(1):1-19, 2016.
-[24] Rawashdeh M.S. and Al-Jammal, H., 2016. Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM. Mediterranean Journal of Mathematics, 13, pp.4661-4677.
+[24] Rawashdeh M.S., Al-Jammal H. Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM. Mediterranean Journal of Mathematics, 13:4661-4677, 2016.
-[25] Debnath, L. and Debnath, L., 2005. Nonlinear partial differential equations for scientists and engineers (pp. 528-529). Boston: Birkhäuser.
+[25] Debnath L. Nonlinear partial differential equations for scientists and engineers. Birkhäuser Boston, MA,  pp. 738, 2005.
-[26] Wazwaz, A.M., 2002. Partial differential equations. CRC Press.
+[26] Wazwaz A.M. Partial differential equations. CRC Press, pp. 476, 2002.
-[27] BULUT, H., 2013. Exact solutions for some fractional nonlinear partial differential equations via Kudryashov method. Physical Sciences, 8(1), pp.24-63.
+[27] Bulut H. Exact solutions for some fractional nonlinear partial differential equations via Kudryashov method. Physical Sciences, 8(1):24-63, 2013.
-[28] Uğurlu, Y. and Kaya, D., 2007. Analytic method for solitary solutions of some partial differential equations. Physics Letters A, 370(3-4), pp.251-259.
+[28] Uğurlu Y., Kaya D. Analytic method for solitary solutions of some partial differential equations. Physics Letters A, 370(3-4):251-259, 2007.
-[29] Odabasi, M. and Misirli, E., 2018. On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences, 41(3), pp.904-911.
+[29] Odabasi M., Misirli E. On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences, 41(3):904-911, 2018.
 [30] Guner, O., Bekir, A. and Ünsal, Ö., 2016. Two reliable methods for solving the time fractional Clannish Random Walker's Parabolic equation. Optik, 127(20), pp.9571-9577.

@@ Line 1,090: / Line 1,090: @@
 [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] Jesus, I.S. and Tenreiro Machado, J.A., 2008. Fractional control of heat diffusion systems. Nonlinear dynamics, 54, pp.263-282.
+[7] Jesus I.S., ¡ Tenreiro Machado J.A. Fractional control of heat diffusion systems. Nonlinear Dynamics, 54:263-282, 2008.
-[8] Feng, Z., 2002. The first-integral method to study the Burgers–Korteweg–de Vries equation. Journal of Physics A: Mathematical and General, 35(2), p.343.
+[8] Feng Z. The first-integral method to study the Burgers–Korteweg–de Vries equation. Journal of Physics A: Mathematical and General, 35(2):343, 2002.
-[9] Ege, S.M. and Misirli, E., 2014. The modified Kudryashov method for solving some fractional-order nonlinear equations. Advances in Difference Equations, 2014, pp.1-13.
+[9] Ege S.M., Misirli E. The modified Kudryashov method for solving some fractional-order nonlinear equations. Advances in Difference Equations, 2014:1-13, 2014.
-[10] Kaplan, M., Bekir, A. and Akbulut, A., 2016. A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dynamics, 85, pp.2843-2850.
+[10] Kaplan M., Bekir A.,  Akbulut A. A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dynamics, 85:2843-2850, 2016.
-[11] Ma, W.X., Wu, H. and He, J., 2007. Partial differential equations possessing Frobenius integrable decompositions. Physics Letters A, 364(1), pp.29-32.
+[11] Ma W.X., Wu H., He J. Partial differential equations possessing Frobenius integrable decompositions. Physics Letters A, 364(1):29-32, 2007.
-[12] Bekir, A. and Güner, Ö., 2014. Analytical approach for the space-time nonlinear partial differential fractional equation. International Journal of Nonlinear Sciences and Numerical Simulation, 15(7-8), pp.463-470.
+[12] Bekir A.,  Güner, Ö. Analytical approach for the space-time nonlinear partial differential fractional equation. International Journal of Nonlinear Sciences and Numerical Simulation, 15(7-8):463-470, 2014.
-[13] Zhang, S. and Zhang, H.Q., 2011. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A, 375(7), pp.1069-1073.
+[13] Zhang S., Zhang H.Q. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A, 375(7):1069-1073, 2011.
-[14] Guo, S., Mei, L., Li, Y. and Sun, Y., 2012. The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics. Physics Letters A, 376(4), pp.407-411.
+[14] Guo S., Mei L., Li Y.,  Sun Y. The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics. Physics Letters A, 376(4):407-411, 2012.
-[15] Jawad, A.J.A.M., Petković, M.D. and Biswas, A., 2010. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation, 217(2), pp.869-877.
+[15] Jawad A.J.A.M., Petković M.D., Biswas A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation, 217(2):869-877, 2010.
-[16] Kilic, B., 2015. Improved šG 0/GŽ-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation. Journal of Computational and Nonlinear Dynamics, 10, pp.051016-1.
+[16] Kilic B. Improved šG 0/GŽ-expansion method for the time-fractional biological population model and Cahn–Hilliard equation. Journal of Computational and Nonlinear Dynamics, 10, pp.051016-1, 2015.
 [17] Kumar, D., Hosseini, K. and Samadani, F., 2017. The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, pp.439-446.
@@ Line 1,122: / Line 1,122: @@
 [22] Sakar, M.G., Erdogan, F. and Yıldırım, A., 2012. Variational iteration method for the time-fractional Fornberg–Whitham equation. Computers & Mathematics with Applications, 63(9), pp.1382-1388.
-[23] Rawashdeh, M.S. and Al-Jammal, H., 2016. New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM. Advances in Difference Equations, 2016(1), pp.1-19.
+[23] Rawashdeh M.S., Al-Jammal H. New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM. Advances in Difference Equations, 2016(1):1-19, 2016.
-[24] Rawashdeh, M.S. and Al-Jammal, H., 2016. Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM. Mediterranean Journal of Mathematics, 13, pp.4661-4677.
+[24] Rawashdeh M.S. and Al-Jammal, H., 2016. Numerical solutions for systems of nonlinear fractional ordinary differential equations using the FNDM. Mediterranean Journal of Mathematics, 13, pp.4661-4677.
 [25] Debnath, L. and Debnath, L., 2005. Nonlinear partial differential equations for scientists and engineers (pp. 528-529). Boston: Birkhäuser.

@@ Line 1,076: / Line 1,076: @@
 ==References==
-[1] Miller, K.S. and Ross, B., 1993. An introduction to the fractional calculus and fractional differential equations.
+<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[2] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., 2006. Theory and applications of fractional differential equations (Vol. 204). elsevier.
+[1] Miller K.S.,  Ross B. An introduction to the fractional calculus and fractional differential equations. 1993
-[3] Kilbas, A.A., Marichev, O.I. and Samko, S.G., 1993. Fractional integrals and derivatives (theory and applications).
+[2] Kilbas A.A., Srivastava H.M.,  Trujillo J.J. Theory and applications of fractional differential equations. (Vol. 204). elsevier, 2006.
-[4] Herzallah, M.A., Muslih, S.I., Baleanu, D. and Rabei, E.M., 2011. Hamilton–Jacobi and fractional like action with time scaling. Nonlinear Dynamics, 66, pp.549-555.
+[3] Kilbas A.A., Marichev O.I.,  Samko S.G. Fractional integrals and derivatives (theory and applications).  1993
-[5] West, B.J., Bologna, M., Grigolini, P., West, B.J., Bologna, M. and Grigolini, P., 2003. Fractional laplace transforms. Physics of Fractal Operators, pp.157-183.
+[4] Herzallah M.A., Muslih S.I., Baleanu D., Rabei E.M. Hamilton–Jacobi and fractional like action with time scaling. Nonlinear Dynamics, 66:549-555, 2011.
-[6] Alqahtani, Z., & Hagag, A. E. (2023). 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), 1-21.
+[5] West B.J., Bologna M., Grigolini P., West B.J., Bologna M., Grigolini P. Fractional laplace transforms. Physics of Fractal Operators, pp. 157-183, 2003.
 [7] Jesus, I.S. and Tenreiro Machado, J.A., 2008. Fractional control of heat diffusion systems. Nonlinear dynamics, 54, pp.263-282.

@@ Line 859: / Line 859: @@
 ==6. Numerical results and discussion==
-For different choices of space and time variables, we conduct numerical simulations for fractional clannish random walker’s parabolic equation of arbitrary order. [[#tab-1|Tables 1]] and [[#tab-2|2]] show the results of a numerical simulation with different values  <math display="inline">x</math> for the initial conditions considered in the first triangular periodic solution and the first multiple soliton-like solution. Similarly, the numerical analysis was carried out for the initial conditions considered in the second triangular periodic solution and the second multiple soliton-like solution, as shown in Tables 3 and 4. Some values for the four cases are also mentioned in [[#tab-1|Tables 1]]-[[#tab-4|4]] at <math display="inline">\epsilon =</math><math>0.95</math> and <math display="inline">\epsilon =0.90</math>. We can deduce from the provided tables that the results generated by FNDM are very accurate. [[#img-1|Figures 1]](a,b), [[#img-2|Figures 2]](a,b), [[#img-3|Figures 3]](a,b), [[#img-4|Figures 4]](a,b) show the behavior of the exact solutions and the NDM solution for four cases. [[#img-1|Figure 1]](c), [[#img-2|Figure 2]](c), [[#img-3|Figure 3]](c) and  [[#img-4|Figure 4]](c)  depict the type of absolute errors for the related equation. [[#img-1|Figure 1]](d), [[#img-2|Figure 2]](d), [[#img-3|Figure 3]](d) and  [[#img-4|Figure 4]](d) depict the graphical representation between exact and NDM solutions for four cases at <math display="inline">\tau =</math><math>0.01</math>. The response of acquired solutions with different values of <math display="inline">\epsilon</math>  between <math display="inline">\varphi \left( x,\tau \right)</math>  and <math display="inline">x</math> is presented in [[#img-1|Figure 1]](e), [[#img-2|Figure 2]](e), [[#img-3|Figure 3]](e) and  [[#img-4|Figure 4]](e). The response of acquired solutions with different values of <math display="inline">\epsilon</math>  between <math display="inline">\varphi \left( x,\tau \right)</math>  and <math display="inline">\tau</math>  is presented in [[#img-1|Figure 1]](f), [[#img-2|Figure 2]](f), [[#img-3|Figure 3]](f) and  [[#img-4|Figure 4]](f). The plots display the dependability and applicability of the predicted technique while analyzing nonlinear issues. Finally, the impact of generalizing models or issues from integer to fractional order can be seen in these plots. Furthermore, numerical simulations have been performed to demonstrate that the proposed technique is viable and effective.
+For different choices of space and time variables, we conduct numerical simulations for fractional clannish random walker’s parabolic equation of arbitrary order. [[#tab-1|Tables 1]] and [[#tab-2|2]] show the results of a numerical simulation with different values  <math display="inline">x</math> for the initial conditions considered in the first triangular periodic solution and the first multiple soliton-like solution. Similarly, the numerical analysis was carried out for the initial conditions considered in the second triangular periodic solution and the second multiple soliton-like solution, as shown in [[#tab-3|Tables 3]] and [[#tab-4|4]]. Some values for the four cases are also mentioned in [[#tab-1|Tables 1]]-[[#tab-4|4]] at <math display="inline">\epsilon =</math><math>0.95</math> and <math display="inline">\epsilon =0.90</math>. We can deduce from the provided tables that the results generated by FNDM are very accurate. [[#img-1|Figures 1]](a,b), [[#img-2|Figures 2]](a,b), [[#img-3|Figures 3]](a,b), [[#img-4|Figures 4]](a,b) show the behavior of the exact solutions and the NDM solution for four cases. [[#img-1|Figure 1]](c), [[#img-2|Figure 2]](c), [[#img-3|Figure 3]](c) and  [[#img-4|Figure 4]](c)  depict the type of absolute errors for the related equation. [[#img-1|Figure 1]](d), [[#img-2|Figure 2]](d), [[#img-3|Figure 3]](d) and  [[#img-4|Figure 4]](d) depict the graphical representation between exact and NDM solutions for four cases at <math display="inline">\tau =</math><math>0.01</math>. The response of acquired solutions with different values of <math display="inline">\epsilon</math>  between <math display="inline">\varphi \left( x,\tau \right)</math>  and <math display="inline">x</math> is presented in [[#img-1|Figure 1]](e), [[#img-2|Figure 2]](e), [[#img-3|Figure 3]](e) and  [[#img-4|Figure 4]](e). The response of acquired solutions with different values of <math display="inline">\epsilon</math>  between <math display="inline">\varphi \left( x,\tau \right)</math>  and <math display="inline">\tau</math>  is presented in [[#img-1|Figure 1]](f), [[#img-2|Figure 2]](f), [[#img-3|Figure 3]](f) and  [[#img-4|Figure 4]](f). The plots display the dependability and applicability of the predicted technique while analyzing nonlinear issues. Finally, the impact of generalizing models or issues from integer to fractional order can be seen in these plots. Furthermore, numerical simulations have been performed to demonstrate that the proposed technique is viable and effective.
 <div id='img-2'></div>

@@ Line 802: / Line 802: @@
 |}
-The approximate solutions and Table 1 demonstrate that the precise solution of Eq. (29) has a generic type that corresponds to the analytical solutions listed above when <math display="inline">\epsilon =1</math>.
+The approximate solutions and [[#tab-1|Table 1]] demonstrate that the precise solution of Eq. (29) has a generic type that corresponds to the analytical solutions listed above when <math display="inline">\epsilon =1</math>.
 ====Second multiple soliton-like solution====