Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation

Adomian decomposition method was used to solve the coupling catalytic reaction model. Coupling catalytic reaction mathematical model is one of the classical models in chemical reaction engineering. Because of its strong nonlinearity, it is difficult to get. Different from the classical two-phase model which contains the transfer coefficients between single phase, the three-mode model contains three transfer coefficients which represent the transfer coefficients of heat and mass between phases and phases [1, 2]. In this case, in order to predict the temperature and fluid concentration distribution in the reactor in more detail, a reduced order three-mode model is needed to analyse the flow problem in the flow passage with homogeneous reaction and wall catalytic reaction in the reactor. Referring to the classical coupling catalytic reaction model, the following dimensionless coupling catalytic reaction model was established: {f(y)∂c∂x=1Pe∂2c∂x2+1p(∂2c∂y2+sy∂c∂y−ϕ2pexp[γhθ1+θ])f(y)∂θ∂x=λLefPe∂2θ∂x2+Lefp(∂2θ∂y2+sy∂θ∂y)+βϕ2pcexp[γhθ1+θ] \left\{{\matrix{{f(y){{\partial c} \over {\partial x}} = {1 \over {Pe}}{{{\partial^2}c} \over {\partial {x^2}}} + {1 \over p}\left({{{{\partial^2}c} \over {\partial {y^2}}} + {s \over y}{{\partial c} \over {\partial y}} - {{{\phi^2}} \over p}\exp \left[ {{{{\gamma_h}\theta} \over {1 + \theta}}} \right]} \right)}\cr{f(y){{\partial \theta} \over {\partial x}} = \lambda {{L{e_f}} \over {Pe}}{{{\partial^2}\theta} \over {\partial {x^2}}} + {{L{e_f}} \over p}\left({{{{\partial^2}\theta} \over {\partial {y^2}}} + {s \over y}{{\partial \theta} \over {\partial y}}} \right) + \beta {{{\phi^2}} \over p}c\exp \left[ {{{{\gamma_h}\theta} \over {1 + \theta}}} \right]}\cr}} \right.

F (y) is the velocity profile. It should be noted that the coupling catalytic reaction has an f coefficient, which has a singularity. The given boundary conditions are as follows: 1Pe∂c∂x=f(y)(c−1),x=0,λLefPe∂θ∂x=f(y)(θ−θin),x=1,∂c∂x=0,∂c∂y=−ϕc2(1+s)cexp[γcθ1+θ],y=1,∂θ∂y=βLefϕc2(1+s)cexp[γcθ1+θ],y=0,∂c∂y=0,y=0∂θ∂y=0,c(0,y)=b0,θ(0,y)=d0. \matrix{{{1 \over {Pe}}{{\partial c} \over {\partial x}} = f(y)(c - 1),}\cr{x = 0,\;\lambda {{L{e_f}} \over {Pe}}{{\partial \theta} \over {\partial x}} = f(y)(\theta- {\theta_{in}}),}\cr{x = 1,\;{{\partial c} \over {\partial x}} = 0,}\cr{{{\partial c} \over {\partial y}} =- {{\phi_c^2} \over {(1 + s)}}c\exp \left[ {{{{\gamma_c}\theta} \over {1 + \theta}}} \right],}\cr{y = 1,\;{{\partial \theta} \over {\partial y}} = {\beta\over {L{e_f}}}{{\phi_c^2} \over {(1 + s)}}c\exp \left[ {{{{\gamma_c}\theta} \over {1 + \theta}}} \right],}\cr{y = 0,\;{{\partial c} \over {\partial y}} = 0,}\cr{y = 0\;{{\partial \theta} \over {\partial y}} = 0,}\cr{c(0,y) = {b_0},}\cr{\theta (0,y) = {d_0}.}\cr}

We define the following linear operators: Lx=∂∂x,Ly=∂∂y,Lxx=∂2∂x2 {L_x} = {\partial\over {\partial x}},{L_y} = {\partial\over {\partial y}},{L_{xx}} = {{{\partial^2}} \over {\partial {x^2}}}

Then it can be deduced that: {f(y)Lxc=1PeLxxc+1p(Lyyc+syLyc)−ϕ2pcN(θ),f(y)Lxθ=λLefPeLxxθ+Lefp(Lyyθ+syLyθ)+βϕ2pcN(θ) \left\{{\matrix{{f(y){L_x}c = {1 \over {Pe}}{L_{xx}}c + {1 \over p}({L_{yy}}c + {s \over y}{L_y}c) - {{{\phi^2}} \over p}cN(\theta),}\cr{f(y){L_x}\theta= \lambda {{L{e_f}} \over {Pe}}{L_{xx}}\theta+ {{L{e_f}} \over p}({L_{yy}}\theta+ {s \over y}{L_y}\theta) + \beta {{{\phi^2}} \over p}cN(\theta)}\cr}} \right.

By operating on the inverse operator, we can continue to get: c=pLyy−1Lxc−pPeLyy−1Lxxc−sLyy−1(1yLyc)+pDaLyy−1[cN(θ)]+A(x)y+B(x),θ=pLefLyy−1Lxθ−λpPeLyy−1Lxxθ−sLyy−1(1yLyθ)−pLefβDaLyy−1[cN(θ)]+C(x)y \matrix{{c = pL_{yy}^{- 1}{L_x}c - {p \over {Pe}}L_{yy}^{- 1}{L_{xx}}c - sL_{yy}^{- 1}({1 \over y}{L_y}c) + pDaL_{yy}^{- 1}[cN(\theta)] + A(x)y + B(x),} \hfill\cr{\theta= {p \over {L{e_f}}}L_{yy}^{- 1}{L_x}\theta- \lambda {p \over {Pe}}L_{yy}^{- 1}{L_{xx}}\theta- sL_{yy}^{- 1}({1 \over y}{L_y}\theta) - {p \over {L{e_f}}}\beta DaL_{yy}^{- 1}[cN(\theta)] + C(x)y} \hfill\cr}

For the recursive algorithm in adomian decomposition method, we use the dual decomposition method: B(x)=∑n=0∞bnxnD(x)=∑n=0∞dnxn \matrix{{B(x) = \sum\limits_{n = 0}^\infty {b_n}{x^n}} \hfill\cr{D(x) = \sum\limits_{n = 0}^\infty {d_n}{x^n}} \hfill\cr}

We use six terms to approximate the exact solution and assume the expression: c¯=c¯(x,y;p,s,λ,u,v,Da,β,γh,b0,b,1…,b5,d0,d1,…,d5)=b0∑n=15cn(x,y;p,s,λ,u,v,Da,β,γh,b0,b1,…,b5,d0,d1,…,d5);θ¯=θ¯(x,y;p,s,λ,u,v,Da,β,γh,b0,b,1…,b5,d0,d1,…,d5)=d0∑n=15θn(x,y;p,s,λ,u,v,Da,β,γh,b0,b1,…,b5,d0,d1,…,d5) \matrix{{\bar c} \hfill & {= \bar c(x,y;p,s,\lambda,u,v,Da,\beta,{\gamma_h},{b_0},{b_,}1 \ldots,{b_5},{d_0},{d_1}, \ldots,{d_5})} \hfill\cr{} \hfill & {= {b_0}\sum\limits_{n = 1}^5{c_n}(x,y;p,s,\lambda,u,v,Da,\beta,{\gamma_h},{b_0},{b_1}, \ldots,{b_5},{d_0},{d_1}, \ldots,{d_5});} \hfill\cr{\bar \theta} \hfill & {= \bar \theta (x,y;p,s,\lambda,u,v,Da,\beta,{\gamma_h},{b_0},{b_,}1 \ldots,{b_5},{d_0},{d_1}, \ldots,{d_5})} \hfill\cr{} \hfill & {= {d_0}\sum\limits_{n = 1}^5{\theta_n}(x,y;p,s,\lambda,u,v,Da,\beta,{\gamma_h},{b_0},{b_1}, \ldots,{b_5},{d_0},{d_1}, \ldots,{d_5})} \hfill\cr}

To study the treatment system of children's audio-visual dysfunction, it is necessary to understand the whole process of audio-visual dysfunction children receiving doctor's training and issuing voice [3, 4]. After asking parents about their child's language diagnosis, the children with audio-visual impairment were evaluated with the use of qiyin bo, and the muscle occlusion of the lower song, lips, tongue and other parts were evaluated, as well as the development of organs W and the sound production. The specific process is shown in Figure 1 below.

Fig. 1

Based on the design of the user experience of users for nuclear chariot, and by observing the demand by the user, we design the product framework, design practice, carry on the usability testing and let the user modify on the test results, thereby reducing development costs. In the process of development, a fixed process is also required. The specific process is shown in Figure 2 below [5, 6].

Fig. 2

Through research concerning the seeing-and-hearing-impaired child's age, we have ascertained that children begin to focus at 2–4 years of age, the age for pre-school children; from the point of view of the user experience, emotional need involves showing people that the product produced in the process of experiencing involves the need to focus on function, demand for visual and sensory experience, and paying attention to the use of communication between process and product. The human experience is realised by the five senses of sight, hearing, sense of smell, touch and taste. The colour survey chart is shown in Figure 3 in detail [7, 8].

Fig. 3

The audio playback function meets the auditory needs of children with audio-visual impairment, as well as the emotional needs of children who are curious about sound. Details of its requirements are shown in Figure 4.

Fig. 4

User requirements have been summarised according to the observation of user behaviour—and the analysis and summary of user functional needs and emotional needs—provided in the previous section. For details of the hierarchical analysis diagram of user requirements, see Figure 5.

Fig. 5

At present, with the development of science and technology, the auxiliary role of APP in treatment is also very prominent. In order to show its auxiliary situation, the required information is presented in the form of a chart; see Figure 6 [9, 10].

Fig. 6