Ecological balance model of effective utilization of agricultural water resources based on fractional differential equations

The water available for farmland irrigation is KG, which is the water available for the Songhua River Water Source Project [5]. The available water quantity of surface water in this irrigation area is defined as KGB. The available water quantity of groundwater is defined as KGX. The relationship between the three is as follows: (1) KG=KGB+KGX KG = KGB + KGX Defining the available surface water KGB is the water volume of the Songhua River at the head of the irrigation area. According to the existing groundwater data in the irrigation area, the available groundwater is defined as the recoverable amount of groundwater determined by the recharge of groundwater.

Under normal circumstances, the recharge of groundwater is equal to the sum of the local precipitation and the groundwater recharge from the side seepage in the front of the mountain. The groundwater replenishment in this area is mainly the replenishment GRB of Songhua River irrigation infiltration [6]. The relationship between the Irrigation Infiltration Replenishment GRB of the Jiseonghua River and the water volume of the Jiseonghua River can be calculated by the following formula: (2) GRB=γ(1−ηchannel)KGB+(1−ηfield)ηchannelKGB GRB = \gamma (1 - {\eta _{{\rm{channel}}}})KGB + (1 - {\eta _{{\rm{field}}}}){\eta _{{\rm{channel}}}}KGB η_channel is the water utilization coefficient of the canal system at all levels in the irrigation area. η_field is the field water utilization coefficient at the end of the irrigation area. γ is the infiltration replenishment coefficient of the canal system at all levels in the irrigation area. The first term at the right end of the formula Eq. (2) is the infiltration replenishment amount of the canal system at all levels in the irrigation area [7]. The second item is the last level of field water infiltration and replenishment in the irrigation area. η_channel is related to the anti-seepage structure of the canals at all levels in the irrigation area. γ is related to the amount of drainage from the last-level canal to the field in the irrigation area. The exploitable coefficient of groundwater is taken as 0.6. The recoverable amount KGX of groundwater is: (3) KGX=0.6(GRB+0.95) KGX = 0.6(GRB + 0.95) Select reasonable parameters for formula (3) according to the existing specific conditions in the irrigation area. To calculate the groundwater recharge of different values from small to large [8]. At the same time, the corresponding mineable amount KGX will also be calculated. Draw the two into Figure 1.

Fig. 1

Due to the circulation of water at the bends of the big river, the sediment transport rate is linked to time to simulate the formation and evolution of river beaches and ridges. In a relatively ideal state, the time-fractional differential equation is used to describe the movement process of quicksand during the formation of the river beach [12]. We give a numerical simulation. It is further organized into a form that is convenient for discussion: (7) (1+2γ2)ηm1−γ2(ηm+11+ηm−11)=γ1(ηm1−p−ηm−p)+ηm0+ρm1 (1 + 2{\gamma _2})\eta _m^1 - {\gamma _2}(\eta _{m + 1}^1 + \eta _{m - 1}^1) = {\gamma _1}(\eta _m^{1 - p} - \eta _m^{ - p}) + \eta _m^0 + \rho _m^1 (8) (1+2γ2)ηmn+1−γ2(ηm+1n+1+ηm−1n+1)=γ1(ηmn+1−p−ηmn−p)+∑k=0n−1(ωk−ωk+1)ηmn−k+ωnηm0+ρmn+1 \left( {1 + 2{\gamma _2}} \right)\eta _m^{n + 1} - {\gamma _2}\left( {\eta _{m + 1}^{n + 1} + \eta _{m - 1}^{n + 1}} \right) = {\gamma _1}\left( {\eta _m^{n + 1 - p} - \eta _m^{n - p}} \right) + \sum\limits_{k = 0}^{n - 1} \left( {{\omega _k} - {\omega _{k + 1}}} \right)\eta _m^{n - k} + {\omega _n}\eta _m^0 + \rho _m^{n + 1} Here, the approximate solution of the different format (7) and formula (8) is η¯mn \overline \eta _m^n , and the error is emn=ηmn−η¯mn e_m^n = \eta _m^n - \overline \eta _m^n .

Prove that ω_k = (k + 1)^1−a − k^1−a, so there is 1=ω0>ω1>⋯>ωk>ωk+1>⋯>0 1 = {\omega _0} > {\omega _1} > \cdots > {\omega _k} > {\omega _{k + 1}} > \cdots > 0 The error equation is obtained from the different format (7) and formula (8): (9) (1+2γ2)εm1−γ2(εm+11+εm−11)=γ1(εm1−p−εm−p)+εm0 (1 + 2{\gamma _2})\varepsilon _m^1 - {\gamma _2}(\varepsilon _{m + 1}^1 + \varepsilon _{m - 1}^1) = {\gamma _1}(\varepsilon _m^{1 - p} - \varepsilon _m^{ - p}) + \varepsilon _m^0 (10) (1+2γ2)εmn+1−γ2(εm+1n+1+εm−1n+1)=γ1(εmn+1−p−εmn−p)+∑k=0n−1(ωk−ωk+1)εmn−k+ωnεm0 (1 + 2{\gamma _2})\varepsilon _m^{n + 1} - {\gamma _2}(\varepsilon _{m + 1}^{n + 1} + \varepsilon _{m - 1}^{n + 1}) = {\gamma _1}(\varepsilon _m^{n + 1 - p} - \varepsilon _m^{n - p}) + \sum\limits_{k = 0}^{n - 1} ({\omega _k} - {\omega _{k + 1}})\varepsilon _m^{n - k} + {\omega _n}\varepsilon _m^0 Let |εjk|=max0≤i≤M|εik| |\varepsilon _j^k| = \mathop {\max }\limits_{0 \le i \le M} |\varepsilon _i^k| , k = 0, 1, 2,⋯ ,N. Then when n = 1, by formula (9): εmn=0 \varepsilon _m^n = 0 , n = −p, −p + 1, ⋯, −1; m = 1, 2, ⋯, M − 1. From equation (9): (11) |εj1|≤|(1+2γ2)εj1−γ2(εj+11+εj−11)|≤γ1|εj1−p−εj−p|+εj0|=(1+2γ1)|εj0| |\varepsilon _j^1| \le |(1 + 2{\gamma _2})\varepsilon _j^1 - {\gamma _2}(\varepsilon _{j + 1}^1 + \varepsilon _{j - 1}^1)| \le {\gamma _1}|\varepsilon _j^{1 - p} - \varepsilon _j^{ - p}| + \varepsilon _j^0| = (1 + 2{\gamma _1})|\varepsilon _j^0| Assuming that when n ≤ k, |εjk|≤(1+2γ1)k|εj0| |\varepsilon _j^k| \le {\left( {1 + 2{\gamma _1}} \right)^k}|\varepsilon _j^0| , is established, when n = k + 1, by formula (11): |εjk+1|≤|(1+2γ1)kεjk+1−γ2(εj+1k+1+εj−1k+1)|≤γ1|εjk+1−p−εjk−p|+∑i=0k−1(ωi−ωi+1)|εjk−i|+ωk|εj0| |\varepsilon _j^{k + 1}| \le |(1 + 2{\gamma _1}{)^k}\varepsilon _j^{k + 1} - {\gamma _2}(\varepsilon _{j + 1}^{k + 1} + \varepsilon _{j - 1}^{k + 1})| \le {\gamma _1}|\varepsilon _j^{k + 1 - p} - \varepsilon _j^{k - p}| + \sum\limits_{i = 0}^{k - 1} ({\omega _i} - {\omega _{i + 1}})|\varepsilon _j^{k - i}| + {\omega _k}|\varepsilon _j^0| Because there is |εjk+1−p|≤|εjk| |\varepsilon _j^{k + 1 - p}| \le |\varepsilon _j^k| , |εjk−p|≤|εjk| |\varepsilon _j^{k - p}| \le |\varepsilon _j^k| , |εjk−i|≤|εjk| |\varepsilon _j^{k - i}| \le |\varepsilon _j^k| , the above formula is less than or equal to 2γ1|εjk|+|εjk|∑i=0k−1(ωi−ωi+1)+ωk|εj0|=2γ1|εjk|+|εjk|(ω0−ωk)+ωk|εj0| 2{\gamma _1}|\varepsilon _j^k| + |\varepsilon _j^k|\sum\limits_{i = 0}^{k - 1} \left( {{\omega _i} - {\omega _{i + 1}}} \right) + {\omega _k}|\varepsilon _j^0| = 2{\gamma _1}|\varepsilon _j^k| + |\varepsilon _j^k|\left( {{\omega _0} - {\omega _k}} \right) + {\omega _k}|\varepsilon _j^0| Taking into account the conditions that ω_k satisfy, the above formula is equal to (12) (1+2γ1)|εjk|≤(1+2γ1)k+1|εj0| (1 + 2{\gamma _1})|\varepsilon _j^k| \le {(1 + 2{\gamma _1})^{k + 1}}|\varepsilon _j^0| Here, the maximum value of k is [TΔt] \left[ {{T \over {\Delta t}}} \right] , which proves that the difference scheme (7) and formula (8) are unconditionally stable. Record the error as εmn=η(xm,tn)−ηmn \varepsilon _m^n = \eta ({x_m},{t_n}) - \eta _m^n here. According to (11): εmn=0 \varepsilon _m^n = 0 , n = −p, −p + 1,⋯ ,−1, 0m = 1, 2,⋯ ,M − 1.

According to formula (7), formula (8) and local truncation error calculation: (13) (1+2γ2)em1−γ2(em+11+em−11)=γ1(em1−p−em−p)+τaΓ(2−a)Rm1 \left( {1 + 2{\gamma _2}} \right)e_m^1 - {\gamma _2}\left( {e_{m + 1}^1 + e_{m - 1}^1} \right) = {\gamma _1}\left( {e_m^{1 - p} - e_m^{ - p}} \right) + {\tau ^a}\Gamma \left( {2 - a} \right)R_m^1 (14) γ1(emn+1−p−emn−p)+∑k=0n−1(ωk−ωk+1)emn−k+τaΓ(2−a)Rmn+1 {\gamma _1}\left( {e_m^{n + 1 - p} - e_m^{n - p}} \right) + \sum\limits_{k = 0}^{n - 1} \left( {{\omega _k} - {\omega _{k + 1}}} \right)e_m^{n - k} + {\tau ^a}\Gamma \left( {2 - a} \right)R_m^{n + 1} Rmn+1 R_m^{n + 1} is the local truncation error. According to Taylor's formula and the integral median theorem, we get: there is a constant C > 0 such that |Rmn|≤C(τ+h2), m=1,2,⋯,M; n=1,2,⋯,N. |R_m^n| \le C(\tau + {h^2}),\quad m = 1,2, \cdots ,M;\quad n = 1,2, \cdots ,N.