October  2021, 14(10): 3351-3386. doi: 10.3934/dcdss.2020440

Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria

1. 

Department of Mathematics and General Sciences, Prince Sultan University-Riyadh-KSA, Saudi Arabia

2. 

Department of Medical Research, China Medical University, Taichung, Taiwan

3. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

4. 

Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran

* Corresponding author: mesamei@gmail.com

Received  October 2019 Revised  February 2020 Published  October 2021 Early access  November 2020

Fund Project: The second author is supported by Bu-Ali Sina University

Crisis intervention in natural disasters is significant to look at from many different slants. In the current study, we investigate the existence of solutions for
$ q $
-integro-differential equation
$ D_q^{\alpha} u(t) + w\left(t , u(t), u'(t), D_q^{\beta} u(t), \int_0^t f(r) u(r) \, {\mathrm d}r, \varphi(u(t)) \right) = 0, $
with three criteria and under some boundary conditions which therein we use the concept of Caputo fractional
$ q $
-derivative and fractional Riemann-Liouville type
$ q $
-integral. New existence results are obtained by applying
$ \alpha $
-admissible map. Lastly, we present two examples illustrating the primary effects.
Citation: Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440
References:
[1]

T. Abdeljawad and J. Alzabut, The $q$-fractional analogue for Gronwall-type inequality, Journal of Function Spaces and Applications, (2013), Art. ID 543839, 7 pp. doi: 10.1155/2013/543839.

[2]

T. Abdeljawad, J. Alzabut and D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, Journal of Inequalities and Applications, (2016), Paper No. 240, 13 pp. doi: 10.1186/s13660-016-1181-2.

[3]

C. Adams, The general theory of a class of linear partial $q$-difference equations, Transactions of the American Mathematical Society, 26 (1924), 283-312.  doi: 10.2307/1989141.

[4]

R. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proceedings of the Cambridge Philosophical Society, 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.

[5]

R. AgarwalD. O'Regan and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 371 (2010), 57-68.  doi: 10.1016/j.jmaa.2010.04.034.

[6]

S. Alizadeh, D. Baleanu and S. Rezapour, Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 55, 19 pp. doi: 10.1186/s13662-020-2527-0.

[7]

R. AlmeidaB. Bastos and M. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[8]

J. Alzabut and T. Abdeljawad, Perron's theorem for $q$-delay difference equations, Applied Mathematics and Information Sciences, 5 (2011), 74-84. 

[9]

M. Annaby and Z. Mansour, $q$-Fractional Calculus and Equations, Springer Heidelberg, 2012. doi: 10.1007/978-3-642-30898-7.

[10]

Z. Bai and T. Qiu, Existence of positive solution for singular fractional differential equation, Applied Mathematics and Computation, 215 (2009), 2761-2767.  doi: 10.1016/j.amc.2009.09.017.

[11]

D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 71, 17 pp. doi: 10.1186/s13662-020-02544-w.

[12]

D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Boundary Value Problems, 2017 (2017), Paper No. 145, 9 pp. doi: 10.1186/s13661-017-0867-9.

[13]

M. Berezowski, Crisis phenomenon in a chemical reactor with recycle, Chemical Engineering Science, 101 (2013), 451-453.  doi: 10.1016/j.ces.2013.07.014.

[14]

A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065.

[15]

R. Carmichael, The general theory of linear $q$-difference equations, American Journal of Mathematics, 34 (1912), 147-168.  doi: 10.2307/2369887.

[16]

R. Ferreira, Nontrivials solutions for fractional $q$-difference boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 70 (2010), 1-101. 

[17]

R. Finkelstein and E. Marcus, Transformation theory of the $q$-oscillator, Journal of Mathematical Physics, 36 (1995), 2652-2672.  doi: 10.1063/1.531057.

[18]

A. GoswamiJ. SinghD. Kumar and Su shila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A: Statistical Mechanics and its Applications, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.

[19]

V. Hedayati and M. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous dirichlet boundary conditions, Boundary Value Problems, 2019 (2019), Paper No. 141, 23 pp. doi: 10.1186/s13661-019-1251-8.

[20]

F. Jackson, $q$-difference equations, American Journal of Mathematics, 32 (1910), 305-314.  doi: 10.2307/2370183.

[21]

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.

[22]

V. Kalvandi and M. E. Samei, New stability results for a sum-type fractional $q$-integro-differential equation, Journal of Advanced Mathematical Studies, 12 (2019), 201-209. 

[23]

A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B. V., Amsterdam, 2006.

[24]

M. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

[25]

S. Liang and M. E. Samei, New approach to solutions of a class of singular fractional $q$-differential problem via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 14, 22 pp. doi: 10.1186/s13662-019-2489-2.

[26]

R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Advances in Difference Equations, 214 (2014), 292, 12 pp. doi: 10.1186/1687-1847-2014-292.

[27] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[28]

S. K. Ntouyas and M. E. Samei, Existence and uniqueness of solutions for multi-term fractional $q$-integro-differential equations via quantum calculus, Advances in Difference Equations, 2019 (2019), Paper No. 475, 20 pp. doi: 10.1186/s13662-019-2414-8.

[29]

P. RajkovićS. Marinković and M. Stanković, Fractional integrals and derivatives in $q$-calculus, Applicable Analysis and Discrete Mathematics, 1 (2007), 311-323. 

[30]

M. E. Samei, Existence of solutions for a system of singular sum fractional $q$-differential equations via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 23, 23 pp. doi: 10.1186/s13662-019-2480-y.

[31]

M. Samei and G. Khalilzadeh Ranjbar, Some theorems of existence of solutions for fractional hybrid $q$-difference inclusion, Journal of Advanced Mathematical Studies, 12 (2019), 63-76. 

[32]

M. E. SameiG. Khalilzadeh Ranjbar and V. Hedayati, Existence of solutions for a class of caputo fractional $q$-difference inclusion on multifunctions by computational results, Kragujevac Journal of Mathematics, 45 (2021), 543-570. 

[33]

M. Samei, V. Hedayati and S. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo–Hadamard type fractional derivative, Advances in Difference Equations, 2019 (2019), Paper No. 163, 15 pp. doi: 10.1186/s13662-019-2090-8.

[34]

M. E. Samei, V. Hedayati and G. K. Ranjbar, The existence of solution for $k$-dimensional system of Langevin Hadamard-type fractional differential inclusions with $2k$ different fractional orders, Mediterranean Journal of Mathematics, 17 (2020), Paper No. 37, 23 pp. doi: 10.1007/s00009-019-1471-2.

[35]

B. SametC. Vetro and P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2154-2165.  doi: 10.1016/j.na.2011.10.014.

[36]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[37]

M. ShabibiM. Postolache and S. Rezapour, Investigation of a multi-singular point-wise defined fractional integro-differential equation, Journal of Mathematical Analysis, 7 (2016), 61-77. 

[38]

M. ShabibiS. Rezapour and S. Vaezpour, A singular fractional integro-differential equation, University Politehnica of Bucharest Scientific Bulletin, Series A, 79 (2017), 109-118. 

[39]

S. Staněk, The existence of positive solutions of singular fractional boundary value problems, Computers & Mathematics with Applications, 62 (2011), 1379-1388.  doi: 10.1016/j.camwa.2011.04.048.

[40]

M. S. Stanković, P. M. Rajković and S. D. Marinković, On $q$-fractional derivatives of Riemann–Liouville and caputo type, C. R. Acad. Bulgare Sci., 63 (2010), 197–-204.

[41]

N. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, Journal of Inequalities and Applications, 2006 (2006), Art. ID 84561, 12 pp. doi: 10.1155/JIA/2006/84561.

[42]

A. Zada, J. Alzabut, H. Waheed and I. L. Popa, Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions, Advances in Difference Equations, 2020 (2020), Paper No. 64, 50 pp. doi: 10.1186/s13662-020-2534-1.

[43]

H. ZhouJ. Alzabut and L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, The European Physical Journal Special Topics, 226 (2017), 3577-3590.  doi: 10.1140/epjst/e2018-00082-0.

show all references

References:
[1]

T. Abdeljawad and J. Alzabut, The $q$-fractional analogue for Gronwall-type inequality, Journal of Function Spaces and Applications, (2013), Art. ID 543839, 7 pp. doi: 10.1155/2013/543839.

[2]

T. Abdeljawad, J. Alzabut and D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, Journal of Inequalities and Applications, (2016), Paper No. 240, 13 pp. doi: 10.1186/s13660-016-1181-2.

[3]

C. Adams, The general theory of a class of linear partial $q$-difference equations, Transactions of the American Mathematical Society, 26 (1924), 283-312.  doi: 10.2307/1989141.

[4]

R. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proceedings of the Cambridge Philosophical Society, 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.

[5]

R. AgarwalD. O'Regan and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 371 (2010), 57-68.  doi: 10.1016/j.jmaa.2010.04.034.

[6]

S. Alizadeh, D. Baleanu and S. Rezapour, Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 55, 19 pp. doi: 10.1186/s13662-020-2527-0.

[7]

R. AlmeidaB. Bastos and M. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[8]

J. Alzabut and T. Abdeljawad, Perron's theorem for $q$-delay difference equations, Applied Mathematics and Information Sciences, 5 (2011), 74-84. 

[9]

M. Annaby and Z. Mansour, $q$-Fractional Calculus and Equations, Springer Heidelberg, 2012. doi: 10.1007/978-3-642-30898-7.

[10]

Z. Bai and T. Qiu, Existence of positive solution for singular fractional differential equation, Applied Mathematics and Computation, 215 (2009), 2761-2767.  doi: 10.1016/j.amc.2009.09.017.

[11]

D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 71, 17 pp. doi: 10.1186/s13662-020-02544-w.

[12]

D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Boundary Value Problems, 2017 (2017), Paper No. 145, 9 pp. doi: 10.1186/s13661-017-0867-9.

[13]

M. Berezowski, Crisis phenomenon in a chemical reactor with recycle, Chemical Engineering Science, 101 (2013), 451-453.  doi: 10.1016/j.ces.2013.07.014.

[14]

A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065.

[15]

R. Carmichael, The general theory of linear $q$-difference equations, American Journal of Mathematics, 34 (1912), 147-168.  doi: 10.2307/2369887.

[16]

R. Ferreira, Nontrivials solutions for fractional $q$-difference boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 70 (2010), 1-101. 

[17]

R. Finkelstein and E. Marcus, Transformation theory of the $q$-oscillator, Journal of Mathematical Physics, 36 (1995), 2652-2672.  doi: 10.1063/1.531057.

[18]

A. GoswamiJ. SinghD. Kumar and Su shila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A: Statistical Mechanics and its Applications, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.

[19]

V. Hedayati and M. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous dirichlet boundary conditions, Boundary Value Problems, 2019 (2019), Paper No. 141, 23 pp. doi: 10.1186/s13661-019-1251-8.

[20]

F. Jackson, $q$-difference equations, American Journal of Mathematics, 32 (1910), 305-314.  doi: 10.2307/2370183.

[21]

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.

[22]

V. Kalvandi and M. E. Samei, New stability results for a sum-type fractional $q$-integro-differential equation, Journal of Advanced Mathematical Studies, 12 (2019), 201-209. 

[23]

A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B. V., Amsterdam, 2006.

[24]

M. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

[25]

S. Liang and M. E. Samei, New approach to solutions of a class of singular fractional $q$-differential problem via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 14, 22 pp. doi: 10.1186/s13662-019-2489-2.

[26]

R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Advances in Difference Equations, 214 (2014), 292, 12 pp. doi: 10.1186/1687-1847-2014-292.

[27] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. 
[28]

S. K. Ntouyas and M. E. Samei, Existence and uniqueness of solutions for multi-term fractional $q$-integro-differential equations via quantum calculus, Advances in Difference Equations, 2019 (2019), Paper No. 475, 20 pp. doi: 10.1186/s13662-019-2414-8.

[29]

P. RajkovićS. Marinković and M. Stanković, Fractional integrals and derivatives in $q$-calculus, Applicable Analysis and Discrete Mathematics, 1 (2007), 311-323. 

[30]

M. E. Samei, Existence of solutions for a system of singular sum fractional $q$-differential equations via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 23, 23 pp. doi: 10.1186/s13662-019-2480-y.

[31]

M. Samei and G. Khalilzadeh Ranjbar, Some theorems of existence of solutions for fractional hybrid $q$-difference inclusion, Journal of Advanced Mathematical Studies, 12 (2019), 63-76. 

[32]

M. E. SameiG. Khalilzadeh Ranjbar and V. Hedayati, Existence of solutions for a class of caputo fractional $q$-difference inclusion on multifunctions by computational results, Kragujevac Journal of Mathematics, 45 (2021), 543-570. 

[33]

M. Samei, V. Hedayati and S. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo–Hadamard type fractional derivative, Advances in Difference Equations, 2019 (2019), Paper No. 163, 15 pp. doi: 10.1186/s13662-019-2090-8.

[34]

M. E. Samei, V. Hedayati and G. K. Ranjbar, The existence of solution for $k$-dimensional system of Langevin Hadamard-type fractional differential inclusions with $2k$ different fractional orders, Mediterranean Journal of Mathematics, 17 (2020), Paper No. 37, 23 pp. doi: 10.1007/s00009-019-1471-2.

[35]

B. SametC. Vetro and P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2154-2165.  doi: 10.1016/j.na.2011.10.014.

[36]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[37]

M. ShabibiM. Postolache and S. Rezapour, Investigation of a multi-singular point-wise defined fractional integro-differential equation, Journal of Mathematical Analysis, 7 (2016), 61-77. 

[38]

M. ShabibiS. Rezapour and S. Vaezpour, A singular fractional integro-differential equation, University Politehnica of Bucharest Scientific Bulletin, Series A, 79 (2017), 109-118. 

[39]

S. Staněk, The existence of positive solutions of singular fractional boundary value problems, Computers & Mathematics with Applications, 62 (2011), 1379-1388.  doi: 10.1016/j.camwa.2011.04.048.

[40]

M. S. Stanković, P. M. Rajković and S. D. Marinković, On $q$-fractional derivatives of Riemann–Liouville and caputo type, C. R. Acad. Bulgare Sci., 63 (2010), 197–-204.

[41]

N. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, Journal of Inequalities and Applications, 2006 (2006), Art. ID 84561, 12 pp. doi: 10.1155/JIA/2006/84561.

[42]

A. Zada, J. Alzabut, H. Waheed and I. L. Popa, Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions, Advances in Difference Equations, 2020 (2020), Paper No. 64, 50 pp. doi: 10.1186/s13662-020-2534-1.

[43]

H. ZhouJ. Alzabut and L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, The European Physical Journal Special Topics, 226 (2017), 3577-3590.  doi: 10.1140/epjst/e2018-00082-0.

Figure 1.  Numerical results of $ A_1(q) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 1
Figure 2.  Numerical results of $ A_2(q) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 2
Figure 3.  Numerical results of $ \Gamma_q(\alpha -1) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 2
Table 1.  Some numerical results for calculation of $ \Gamma_q(x) $ with $ q = \frac{1}{8} $ which is constant, for $ x = 9.5, 65,110,780 $ in Algorithm 2
$ n $ $ x=9.5 $ $ x=65 $ $ x=110 $ $ x=780 $
$ 1 $ $ 2.679786 $ $ 4432.545834 $ $ 1804225.634753 $ $ 1.29090809480473E+45 $
$ 2 $ $ 2.674552 $ $ 4423.888518 $ $ 1800701.756560 $ $ 1.28838678993206E+45 $
$ 3 $ $ 2.673899 $ $ 4422.808467 $ $ 1800262.132108 $ $ 1.28807224237593E+45 $
$ 4 $ $ 2.673818 $ $ 4422.673494 $ $ 1800207.192468 $ $ 1.28803293353064E+45 $
$ 5 $ $ 2.673808 $ $ 4422.656623 $ $ 1800200.325222 $ $ 1.28802802007493E+45 $
$ 6 $ 2.673806 $ 4422.654514 $ $ 1800199.466820 $ $ 1.28802740589531E+45 $
$ 7 $ $ 2.673806 $ $ 4422.654250 $ $ 1800199.359519 $ $ 1.28802732912289E+45 $
$ 8 $ $ 2.673806 $ $ 4422.654217 $ $ 1800199.346107 $ $ 1.28802731952634E+45 $
$ 9 $ $ 2.673806 $ $ 4422.654213 $ $ 1800199.344430 $ $ 1.28802731832677E+45 $
$ 10 $ $ 2.673806 $ $ 4422.654213 $ $ 1800199.344221 $ $ 1.28802731817683E+45 $
$ 11 $ $ 2.673806 $ 4422.654212 $ 1800199.344195 $ $ 1.28802731815808E+45 $
$ 12 $ $ 2.673806 $ $ 4422.654212 $ 1800199.344191 $ 1.28802731815574E+45 $
$ 13 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815545E+45 $
$ 14 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ 1.28802731815541E+45
$ 15 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 16 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 17 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 18 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 19 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ n $ $ x=9.5 $ $ x=65 $ $ x=110 $ $ x=780 $
$ 1 $ $ 2.679786 $ $ 4432.545834 $ $ 1804225.634753 $ $ 1.29090809480473E+45 $
$ 2 $ $ 2.674552 $ $ 4423.888518 $ $ 1800701.756560 $ $ 1.28838678993206E+45 $
$ 3 $ $ 2.673899 $ $ 4422.808467 $ $ 1800262.132108 $ $ 1.28807224237593E+45 $
$ 4 $ $ 2.673818 $ $ 4422.673494 $ $ 1800207.192468 $ $ 1.28803293353064E+45 $
$ 5 $ $ 2.673808 $ $ 4422.656623 $ $ 1800200.325222 $ $ 1.28802802007493E+45 $
$ 6 $ 2.673806 $ 4422.654514 $ $ 1800199.466820 $ $ 1.28802740589531E+45 $
$ 7 $ $ 2.673806 $ $ 4422.654250 $ $ 1800199.359519 $ $ 1.28802732912289E+45 $
$ 8 $ $ 2.673806 $ $ 4422.654217 $ $ 1800199.346107 $ $ 1.28802731952634E+45 $
$ 9 $ $ 2.673806 $ $ 4422.654213 $ $ 1800199.344430 $ $ 1.28802731832677E+45 $
$ 10 $ $ 2.673806 $ $ 4422.654213 $ $ 1800199.344221 $ $ 1.28802731817683E+45 $
$ 11 $ $ 2.673806 $ 4422.654212 $ 1800199.344195 $ $ 1.28802731815808E+45 $
$ 12 $ $ 2.673806 $ $ 4422.654212 $ 1800199.344191 $ 1.28802731815574E+45 $
$ 13 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815545E+45 $
$ 14 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ 1.28802731815541E+45
$ 15 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 16 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 17 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 18 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
$ 19 $ $ 2.673806 $ $ 4422.654212 $ $ 1800199.344191 $ $ 1.28802731815541E+45 $
Table 2.  Some numerical results for calculation of $ \Gamma_q(x) $ with $ q = \frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9} $ for $ x = 9.5 $ of Algorithm 2
$ n $ $ q=\frac{1}{8} $ $ q=\frac{1}{2} $ $ q=\frac{4}{5} $ $ q=\frac{8}{9} $
$ 1 $ $ 2.679786 $ $ 136.046206 $ $ 79062.138227 $ $ 6301918.338883 $
$ 2 $ $ 2.674552 $ $ 119.081545 $ $ 41793.335091 $ $ 2528395.395827 $
$ 3 $ $ 2.673899 $ $ 111.658224 $ $ 26290.733638 $ $ 1232715.590371 $
$ 4 $ $ 2.673818 $ $ 108.178242 $ $ 18589.881264 $ $ 689176.848061 $
$ 5 $ $ 2.673808 $ $ 106.492553 $ $ 14278.326587 $ $ 426538.394173 $
$ 6 $ 2.673806 $ 105.662861 $ $ 11650.586796 $ $ 285518.687713 $
$ 7 $ $ 2.673806 $ $ 105.251251 $ $ 9946.3508930 $ $ 203363.796571 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 26 $ $ 2.673806 $ $ 104.841780 $ $ 5522.283831 $ $ 25842.863721 $
$ 27 $ $ 2.673806 $ $ 104.841780 $ $ 5513.202433 $ $ 25230.371788 $
$ 28 $ $ 2.673806 $ 104.841779 $ 5505.949683 $ $ 24699.649904 $
$ 29 $ $ 2.673806 $ $ 104.841779 $ $ 5500.155385 $ $ 24238.446645 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 106 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048235 $ $ 20879.606269 $
$ 107 $ $ 2.673806 $ $ 104.841779 $ 5477.048234 $ 20879.566792 $
$ 108 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.531702 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 118 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.337427 $
$ 119 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.327822 $
$ 120 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ 20879.319284
$ n $ $ q=\frac{1}{8} $ $ q=\frac{1}{2} $ $ q=\frac{4}{5} $ $ q=\frac{8}{9} $
$ 1 $ $ 2.679786 $ $ 136.046206 $ $ 79062.138227 $ $ 6301918.338883 $
$ 2 $ $ 2.674552 $ $ 119.081545 $ $ 41793.335091 $ $ 2528395.395827 $
$ 3 $ $ 2.673899 $ $ 111.658224 $ $ 26290.733638 $ $ 1232715.590371 $
$ 4 $ $ 2.673818 $ $ 108.178242 $ $ 18589.881264 $ $ 689176.848061 $
$ 5 $ $ 2.673808 $ $ 106.492553 $ $ 14278.326587 $ $ 426538.394173 $
$ 6 $ 2.673806 $ 105.662861 $ $ 11650.586796 $ $ 285518.687713 $
$ 7 $ $ 2.673806 $ $ 105.251251 $ $ 9946.3508930 $ $ 203363.796571 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 26 $ $ 2.673806 $ $ 104.841780 $ $ 5522.283831 $ $ 25842.863721 $
$ 27 $ $ 2.673806 $ $ 104.841780 $ $ 5513.202433 $ $ 25230.371788 $
$ 28 $ $ 2.673806 $ 104.841779 $ 5505.949683 $ $ 24699.649904 $
$ 29 $ $ 2.673806 $ $ 104.841779 $ $ 5500.155385 $ $ 24238.446645 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 106 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048235 $ $ 20879.606269 $
$ 107 $ $ 2.673806 $ $ 104.841779 $ 5477.048234 $ 20879.566792 $
$ 108 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.531702 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 118 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.337427 $
$ 119 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ $ 20879.327822 $
$ 120 $ $ 2.673806 $ $ 104.841779 $ $ 5477.048234 $ 20879.319284
Table 3.  Some numerical results for calculation of $ \Gamma_q(x) $ with $ q = \frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9} $ for $ x = 110 $ of Algorithm 2
$ n $ $ q=\frac{1}{8} $ $ q=\frac{1}{2} $ $ q=\frac{4}{5} $ $ q=\frac{8}{9} $
$ 1 $ $ 1804225.634753 $ $ 2.43388915243820E+32 $ $ 1.10933564801075E+75 $ $ 2.3996994906237E+102 $
$ 2 $ $ 1800701.756560 $ $ 2.12965300838343E+32 $ $ 5.41355796236824E+74 $ $ 7.1431517307455E+101 $
$ 3 $ $ 1800262.132108 $ $ 1.99654969535946E+32 $ $ 3.19616462101800E+74 $ $ 2.6837217226512E+101 $
$ 4 $ $ 1800207.192468 $ $ 1.93415751737948E+32 $ $ 2.14884539802207E+74 $ $ 1.1944485864825E+101 $
$ 5 $ $ 1800200.325222 $ $ 1.90393630617042E+32 $ $ 1.58553847001434E+74 $ $ 6.0526350536381E+100 $
$ 6 $ $ 1800199.466820 $ $ 1.88906180377847E+32 $ $ 1.25302695267477E+74 $ $ 3.3987862057282E+100 $
$ 7 $ $ 1800199.359519 $ $ 1.88168265610746E+32 $ $ 1.04280391429109E+74 $ $ 2.0741306563269E+100 $
$ 8 $ $ 1800199.346107 $ $ 1.87800749466975E+32 $ $ 9.02841142168746E+73 $ $ 1.3555712905453E+100 $
$ 9 $ $ 1800199.344430 $ $ 1.87617350297573E+32 $ $ 8.05899312693661E+73 $ $ 9.38129101307050E+99 $
$ 10 $ $ 1800199.344221 $ $ 1.87525740263248E+32 $ $ 7.36673088857628E+73 $ $ 6.81335603265770E+99 $
$ 11 $ $ 1800199.344195 $ $ 1.87479957611817E+32 $ $ 6.86049299667128E+73 $ $ 5.15556440821410E+99 $
$ 12 $ 1800199.344191 $ 1.87457071874804E+32 $ $ 6.48333340557523E+73 $ $ 4.04051908444650E+99 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 48 $ $ 1800199.344191 $ 1.87434189862553E+32 $ 5.18960499065178E+73 $ $ 6.66324790738213E+98 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 90 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ 5.18923469131315E+73 $ 6.50025876524830E+98 $
$ 91 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923468501255E+73 $ $ 6.50013085733126E+98 $
$ 92 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923467997207E+73 $ $ 6.50001716364224E+98 $
$ 93 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923467593968E+73 $ $ 6.49991610435300E+98 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 118 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465987107E+73 $ $ 6.49915022957670E+98 $
$ 119 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465985889E+73 $ $ 6.49914550293450E+98 $
$ 120 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465984914E+73 $ 6.49914130147782E+98
$ n $ $ q=\frac{1}{8} $ $ q=\frac{1}{2} $ $ q=\frac{4}{5} $ $ q=\frac{8}{9} $
$ 1 $ $ 1804225.634753 $ $ 2.43388915243820E+32 $ $ 1.10933564801075E+75 $ $ 2.3996994906237E+102 $
$ 2 $ $ 1800701.756560 $ $ 2.12965300838343E+32 $ $ 5.41355796236824E+74 $ $ 7.1431517307455E+101 $
$ 3 $ $ 1800262.132108 $ $ 1.99654969535946E+32 $ $ 3.19616462101800E+74 $ $ 2.6837217226512E+101 $
$ 4 $ $ 1800207.192468 $ $ 1.93415751737948E+32 $ $ 2.14884539802207E+74 $ $ 1.1944485864825E+101 $
$ 5 $ $ 1800200.325222 $ $ 1.90393630617042E+32 $ $ 1.58553847001434E+74 $ $ 6.0526350536381E+100 $
$ 6 $ $ 1800199.466820 $ $ 1.88906180377847E+32 $ $ 1.25302695267477E+74 $ $ 3.3987862057282E+100 $
$ 7 $ $ 1800199.359519 $ $ 1.88168265610746E+32 $ $ 1.04280391429109E+74 $ $ 2.0741306563269E+100 $
$ 8 $ $ 1800199.346107 $ $ 1.87800749466975E+32 $ $ 9.02841142168746E+73 $ $ 1.3555712905453E+100 $
$ 9 $ $ 1800199.344430 $ $ 1.87617350297573E+32 $ $ 8.05899312693661E+73 $ $ 9.38129101307050E+99 $
$ 10 $ $ 1800199.344221 $ $ 1.87525740263248E+32 $ $ 7.36673088857628E+73 $ $ 6.81335603265770E+99 $
$ 11 $ $ 1800199.344195 $ $ 1.87479957611817E+32 $ $ 6.86049299667128E+73 $ $ 5.15556440821410E+99 $
$ 12 $ 1800199.344191 $ 1.87457071874804E+32 $ $ 6.48333340557523E+73 $ $ 4.04051908444650E+99 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 48 $ $ 1800199.344191 $ 1.87434189862553E+32 $ 5.18960499065178E+73 $ $ 6.66324790738213E+98 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 90 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ 5.18923469131315E+73 $ 6.50025876524830E+98 $
$ 91 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923468501255E+73 $ $ 6.50013085733126E+98 $
$ 92 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923467997207E+73 $ $ 6.50001716364224E+98 $
$ 93 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923467593968E+73 $ $ 6.49991610435300E+98 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 118 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465987107E+73 $ $ 6.49915022957670E+98 $
$ 119 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465985889E+73 $ $ 6.49914550293450E+98 $
$ 120 $ $ 1800199.344191 $ $ 1.87434189862553E+32 $ $ 5.18923465984914E+73 $ 6.49914130147782E+98
Table 4.  Some numerical results for calculation of $\ell$, $\ell'$ and $A_1(q)$ in Example 1 for $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$
$ n $ $ q = \frac{1}{7} $ $ q = \frac{1}{2} $ $ q = \frac{8}{9} $
$ \ell, \ell' $ $ A_1(q) $ $ \ell, \ell' $ $ A_1(q) $ $ \ell, \ell' $ $ A_1(q) $
$ 1 $ $ 1.0124 $ $ 0 $ $ 0.9946 $ $ 0 $ $ 0.8133 $ $ 0 $
$ 2 $ $ 1.0132 $ $ 0.1243 $ $ 1.0132 $ $ 0.2154 $ $ 0.8462 $ $ 0.0164 $
$ 3 $ $ 1.0133 $ $ 0.1406 $ $ 1.0224 $ $ 0.3091 $ $ 0.873 $ $ 0.0376 $
$ 4 $ $ 1.0133 $ $ 0.1429 $ $ 1.0269 $ $ 0.3539 $ $ 0.8954 $ $ 0.0621 $
$ 5 $ 1.0133 0.1433 $ 1.0291 $ $ 0.3759 $ $ 0.9143 $ $ 0.0888 $
$ 6 $ $ 1.0133 $ $ 0.1433 $ $ 1.0302 $ $ 0.3868 $ $ 0.9305 $ $ 0.1165 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 13 $ $ 1.0133 $ $ 0.1433 $ $ 1.0313 $ $ 0.3975 $ $ 0.9972 $ $ 0.2933 $
$ 14 $ $ 1.0133 $ $ 0.1433 $ 1.0314 0.3976 $ 1.0026 $ $ 0.313 $
$ 15 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0073 $ $ 0.3312 $
$ 16 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0115 $ $ 0.3478 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 80 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4985 $
$ 81 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4985 $
$ 82 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ 1.0442 0.4986
$ 83 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4986 $
$ n $ $ q = \frac{1}{7} $ $ q = \frac{1}{2} $ $ q = \frac{8}{9} $
$ \ell, \ell' $ $ A_1(q) $ $ \ell, \ell' $ $ A_1(q) $ $ \ell, \ell' $ $ A_1(q) $
$ 1 $ $ 1.0124 $ $ 0 $ $ 0.9946 $ $ 0 $ $ 0.8133 $ $ 0 $
$ 2 $ $ 1.0132 $ $ 0.1243 $ $ 1.0132 $ $ 0.2154 $ $ 0.8462 $ $ 0.0164 $
$ 3 $ $ 1.0133 $ $ 0.1406 $ $ 1.0224 $ $ 0.3091 $ $ 0.873 $ $ 0.0376 $
$ 4 $ $ 1.0133 $ $ 0.1429 $ $ 1.0269 $ $ 0.3539 $ $ 0.8954 $ $ 0.0621 $
$ 5 $ 1.0133 0.1433 $ 1.0291 $ $ 0.3759 $ $ 0.9143 $ $ 0.0888 $
$ 6 $ $ 1.0133 $ $ 0.1433 $ $ 1.0302 $ $ 0.3868 $ $ 0.9305 $ $ 0.1165 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 13 $ $ 1.0133 $ $ 0.1433 $ $ 1.0313 $ $ 0.3975 $ $ 0.9972 $ $ 0.2933 $
$ 14 $ $ 1.0133 $ $ 0.1433 $ 1.0314 0.3976 $ 1.0026 $ $ 0.313 $
$ 15 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0073 $ $ 0.3312 $
$ 16 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0115 $ $ 0.3478 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 80 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4985 $
$ 81 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4985 $
$ 82 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ 1.0442 0.4986
$ 83 $ $ 1.0133 $ $ 0.1433 $ $ 1.0314 $ $ 0.3976 $ $ 1.0442 $ $ 0.4986 $
Table 5.  Some numerical results for calculation of $A_3(q)$ in Example 2 for $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$. One can verify that $A_3(q) < \Gamma_q(\alpha-1)$ for all $n$ when $q$ changes
$ n $ $ q = \frac{1}{7} $ $ q = \frac{1}{2} $ $ q = \frac{8}{9} $
$ A_3(q) $ $ \Gamma_q(\alpha-1) $ $ A_3(q) $ $ \Gamma_q(\alpha-1) $ $ A_3(q) $ $ \Gamma_q(\alpha-1) $
$ 1 $ $ 1.3246 $ $ 1.6802 $ $ 1.3085 $ $ 8.774 $ $ 1.393 $ $ 1799.5494 $
$ 2 $ $ 1.2731 $ $ 1.6753 $ $ 1.1294 $ $ 7.7199 $ $ 1.0942 $ $ 913.1535 $
$ 3 $ 1.2666 1.6746 $ 1.0755 $ $ 7.2574 $ $ 0.9855 $ $ 542.374 $
$ 4 $ $ 1.2656 $ $ 1.6745 $ $ 1.0539 $ $ 7.0403 $ $ 0.9363 $ $ 358.4859 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 19 $ $ 1.2655 $ $ 1.6745 $ $ 1.0350 $ $ 6.8321 $ $ 0.8635 $ $ 46.4785 $
$ 20 $ $ 1.2655 $ $ 1.6745 $ 1.0350 6.832 $ 0.8632 $ $ 44.7825 $
$ 21 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8629 $ $ 43.3383 $
$ 22 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8627 $ $ 42.1023 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 113 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6244 $
$ 114 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ 0.8612 33.6243
$ 115 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6243 $
$ 116 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6243 $
$ n $ $ q = \frac{1}{7} $ $ q = \frac{1}{2} $ $ q = \frac{8}{9} $
$ A_3(q) $ $ \Gamma_q(\alpha-1) $ $ A_3(q) $ $ \Gamma_q(\alpha-1) $ $ A_3(q) $ $ \Gamma_q(\alpha-1) $
$ 1 $ $ 1.3246 $ $ 1.6802 $ $ 1.3085 $ $ 8.774 $ $ 1.393 $ $ 1799.5494 $
$ 2 $ $ 1.2731 $ $ 1.6753 $ $ 1.1294 $ $ 7.7199 $ $ 1.0942 $ $ 913.1535 $
$ 3 $ 1.2666 1.6746 $ 1.0755 $ $ 7.2574 $ $ 0.9855 $ $ 542.374 $
$ 4 $ $ 1.2656 $ $ 1.6745 $ $ 1.0539 $ $ 7.0403 $ $ 0.9363 $ $ 358.4859 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 19 $ $ 1.2655 $ $ 1.6745 $ $ 1.0350 $ $ 6.8321 $ $ 0.8635 $ $ 46.4785 $
$ 20 $ $ 1.2655 $ $ 1.6745 $ 1.0350 6.832 $ 0.8632 $ $ 44.7825 $
$ 21 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8629 $ $ 43.3383 $
$ 22 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8627 $ $ 42.1023 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 113 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6244 $
$ 114 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ 0.8612 33.6243
$ 115 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6243 $
$ 116 $ $ 1.2655 $ $ 1.6745 $ $ 1.035 $ $ 6.832 $ $ 0.8612 $ $ 33.6243 $
[1]

Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 551-568. doi: 10.3934/naco.2021021

[2]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[3]

Pak Tung Ho. Prescribing the $ Q' $-curvature in three dimension. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2285-2294. doi: 10.3934/dcds.2019096

[4]

Gilbert Strang. Three steps on an open road. Inverse Problems and Imaging, 2013, 7 (3) : 961-966. doi: 10.3934/ipi.2013.7.961

[5]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885

[6]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[7]

Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030

[8]

Gustavo S. Costa, Giovany M. Figueiredo. Existence and concentration of nodal solutions for a subcritical p&q equation. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5077-5095. doi: 10.3934/cpaa.2020227

[9]

Chungen Liu, Yafang Wang. Existence results for the fractional Q-curvature problem on three dimensional CR sphere. Communications on Pure and Applied Analysis, 2018, 17 (3) : 849-885. doi: 10.3934/cpaa.2018043

[10]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[11]

Gokhan Yener, Ibrahim Emiroglu. A q-analogue of the multiplicative calculus: Q-multiplicative calculus. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1435-1450. doi: 10.3934/dcdss.2015.8.1435

[12]

Harman Kaur, Meenakshi Rana. Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $. Electronic Research Archive, 2021, 29 (6) : 4257-4268. doi: 10.3934/era.2021084

[13]

Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037

[14]

Jose S. Cánovas. On q-deformed logistic maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2833-2848. doi: 10.3934/dcdsb.2021162

[15]

Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217

[16]

Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483

[17]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090

[18]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217

[19]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537

[20]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (297)
  • HTML views (367)
  • Cited by (0)

Other articles
by authors

[Back to Top]