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

Thabet Abdeljawad; Mohammad Esmael Samei

doi:10.3934/dcdss.2020440

Article Contents

2021, Volume 14, Issue 10: 3351-3386. Doi: 10.3934/dcdss.2020440

This issue Previous Article Oscillation criteria for kernel function dependent fractional dynamic equations Next Article Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator

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

Thabet Abdeljawad^1,2,3, and
Mohammad Esmael Samei^4, ,

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
^* Corresponding author: mesamei@gmail.com

Received: September 30, 2019

Revised: January 31, 2020

Early access: November 2020

Published: October 2021

The second author is supported by Bu-Ali Sina University

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34A08, 34B16; Secondary: 39A13.

Citation:

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Figure 1. Numerical results of $ A_1(q) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 1

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Figure 2. Numerical results of $ A_2(q) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 2

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Figure 3. Numerical results of $ \Gamma_q(\alpha -1) $ where $ q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9} $ in Example 2

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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 $

| Show Table

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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

| Show Table

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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

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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} $
$ n $	$ \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 $

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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} $
$ n $	$ 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 $

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