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

• * Corresponding author: mesamei@gmail.com

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.

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

 Citation: • • 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 $15$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $16$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $17$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $18$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $19$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$

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

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

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 $83$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4986$

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 $115$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$ $116$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$
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