Financial accounting measurement model based on numerical analysis of rigid normal differential equation and rigid generalised functional equation

With the expansion of the application field of the differential equation, it becomes very important to study the solution of the differential equation. In recent years, many famous mathematicians in the world have made more in-depth and extensive research on rigid differential equations and rigid generalised functional equations by analysing theories and methods. The solution of the differential equation not only contains a component attenuation that is very quick and contains a relatively slow weight change, when they try to change in the solution of the slow interval value to solve these problems, even as attenuation of the fast varying component values are insignificant, but also the disturbance of rapid change still seriously affects the stability and accuracy of the numerical solution, creating a lot of real difficulties for the calculation. These processes are called ‘rigid’, and the initial value problems of ordinary differential equations describing these processes are called ‘rigid’.

The research on rigid differential equations has been conducted for >200 years, and the research object is mainly in the pure theory field of financial accounting. With the widespread application of differential equations and generalised functional equations, the research process has gradually attracted the attention of the general financial accounting field. In recent decades, differential equations have achieved breakthrough development in the fields of biology, rheology, chemical data processing, signal processing identification, control theory and financial accounting. Their main principle is to convert them into equivalent integral equations using Green's function, and when the nonlinear term satisfies certain conditions, we use the property of noncompactness measure and the fixed point theorem to prove the related problems of the solution, and the main idea used is the idea of transformation. The new results are of great significance in the theory and practical application of financial accounting. Differential boundary value problem is one of the most active research fields in analytical mathematics, which originated from various applied disciplines. A large number of physical models and natural phenomena can be simplified to solve the boundary value problem of the equation. Therefore, it is particularly important to study the boundary value problems of differential equations and generalised functional equations [1].

Researchers first coined the term ‘rigid’, and they were the first to come up with a class of numerical methods known as the BDF method. It is important to note that the BDF method they proposed was very mature at an early stage and is still an essential part of many highly successful numerical methods for some rigid problems. Since 1952, numerical methods for rigid problems have been widely used in the field of financial accounting, especially in the last two decades. Hundreds’ paper not only solves the problem of constructing efficient algorithms for financial accounting measurement but also provides financial accounting theoretical analysis for these algorithms. A large number of remarkable results have been obtained in the study of rigid differential equation algorithm theory. The early literature on rigidity focused on the linear stability of numerical algorithms in financial accounting, that is, the numerical stability of the scalar model equation Y’ = λ Y, (λ ∈ C) is studied. The related stability concepts include A – stability, A (α) – stability and rigid stability, etc. These theories and concepts based on linear model equations have been successfully used to guide more complex numerical problems, but from a strict mathematical point of view, these theories can only be applied to linear autonomous systems with constant coefficients, and cannot be used as the theoretical basis for studying the stability of numerical methods for rigid nonlinear problems. Moreover, the convergence analysis of numerical algorithms for rigid problems is even neglected in the early literature.

Two typical methods are the backward differential formula and the Runge-Kutta method when dealing with rigid problems. Backward differential formulas are particularly applicable to rigid differential equations and differential algebraic equations. The earliest methods of BDF can be traced back to 1952 by Yan Y and Loureno I although the name of the BDF method was not given at that time. However, for the rigid problem, the BDF method shows outstanding stability properties that allow larger step sizes compared with the explicit method [2,3]. Especially since Badri gave a classical introduction in 1971, the BDF method has attracted more attention from many scholars. Badri M proposed that the first and second-order methods of backward differential formula meet the A-stability and L-stability conditions. All methods of order 3 to order 6 satisfy the weaker A (α)-stability and rigid stability conditions, and there is no available BDF formula higher than order 6 [4].

And all the formulas in BDF format are extremely stable at infinity. Although the convergence order to the BDF method is limited, it has outstanding advantages, namely high practical calculation efficiency, extreme stability at the point of infinity leads to the rapid attenuation of the error of the rigid component. This makes the BDF method known as the numerical method that has long been used to solve the rigid problem. For example, Xia Y Q et al. constructed A cyclic linear method that satisfies A (α)-stability from order 1 to order 7 in 1978, which is more stable than the BDF method [5]. In addition, the EBDF method proposed by Portioli F in 1980 and the modified EBDF method proposed by Portioli F in 1983, the methods of order 2 to order 4 are A-stable, and the methods of order 5 to order 9 are A (α)-stable. Its stability is better than that of the BDF method, and its disadvantages are, in other words, the actual calculation amount of each step is larger than BDF, but the overall effect is better than BDF [6]. Swann J D systematically studied the stability of rigid V Olterra type functional differential equations and the shrinkage, asymptotic stability, B-stability and B-convergence of Runge-Kutta method and general linear method [7]. Chang M et al. obtained the conditions of stability and asymptotic stability of the Runge-Kutta method for solving nonlinear neutral delay integral and differential equations [8]. Plaksiy et al. solved the Runge-Kutta method, Pouzet-Runge-Kutta method and the asymptotic stability of the single-branch method for a class of nonlinear neutral delay integral-differential equations [9]. Koriga S et al. solved the stability of numerical solutions of several types of nonlinear neutral functional integral and differential equations [10].

Let X be a real or complex Banach space and ‖•‖ denote the norm therein. For any closed interval, the symbol C_X (I) represents the Banach space formed by all continuous mappings x : I → X, and the norm is defined as ‖x‖∞=maxt∈I‖x(t)‖ {\left\| x \right\|_\infty } = \mathop {\max }\limits_{t \in I} \left\| {x(t)} \right\| . Consider the initial value problem (1) {y′(t)=f(t,y(t),y(⋅)),0≤t≤T,y(t)=ϕ(t), −τ≤t≤0, \left\{ {\matrix{ {y'(t) = f(t,y(t),y( \cdot )),0 \le t \le T,} \hfill \cr {y(t) = \phi (t),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \tau \le t \le 0,} \hfill \cr } } \right. Here T τ is a real constant, T > 0, τ ≥ 0, φ ∈ C_X [−τ, 0], is a given initial function, f : [0, T]× X × C_X [−τ, T] → X is a given continuous mapping that satisfies this condition (2) {(1−α(t)λ)Gf(0,t,u,v,ψ)≤Gf(λ,t,u,v,ψ)∀λ≥0,t∈[0,T],u,v∈X,ψ∈CX[−τ,T],‖f(t,u,ψ1)−f(t,u,ψ2)‖≤β(t)maxt−μ2(t)≤ξ≤t−μ1(t)‖ψ1(ξ)−ψ2(ξ)‖∀t∈[0,T],u∈X,ψ1,ψ2∈CX[−τ,T], \left\{ {\matrix{ {(1 - \alpha (t)\lambda ){G_f}(0,t,u,v,\psi ) \le {G_f}(\lambda ,t,u,v,\psi )} \hfill \cr {\forall \lambda \ge 0,t \in \left[ {0,T} \right],u,v \in X,\psi \in {C_X}\left[ { - \tau ,T} \right],} \hfill \cr {\left\| {f(t,u,{\psi _1}) - f(t,u,{\psi _2})} \right\| \le \beta (t)\mathop {\max }\limits_{t - {\mu _2}(t) \le \xi \le t - {\mu _1}(t)} \left\| {{\psi _1}(\xi ) - {\psi _2}(\xi )} \right\|} \hfill \cr {\forall t \in \left[ {0,T} \right],u \in X,{\psi _1},{\psi _2} \in {C_X}\left[ { - \tau ,T} \right],} \hfill \cr } } \right. where α(t) and β(t) are continuous functions, the functions μ₁(t) and μ₂(t) satisfy the inequality 0 ≤ μ₁(t) ≤ μ₂(t) ≤ t + τ ∀t ∈ [0, T].

Note that for the special case where X is a Hilbert space, condition (2a) is equivalent to the one-sided Lipschitz condition. We assume that there exists a true solution y(t), 0 ≤ t ≤ T for problem (1), and denoted by z(t) the solution of the perturbation problem obtained by replacing the initial function φ(t) with ψ(t) in problem (1). We obtained the following results from the paper.

That's true, right there c=[sup0≤t≤T(α(t)+β(t))]+ c = {\left[ {\mathop {\sup }\limits_{0 \le t \le T} (\alpha (t) + \beta (t))} \right]_ + } Here and hereafter, we define [x]+={xx>0,0x≤0. {\left[ x \right]_ + } = \left\{ {\matrix{ {x} & {x > 0,} \hfill \cr {0}& {x \le 0.} \hfill \cr } } \right.

The following two theorems apply to the case where the integral interval of problem (1) is [0,+∞.

Here, {Cμ=v+(1−v)exp(α0μ)∈(0,1),v=sup0≤t≤+∞(β(t)/|α(t)|)<1,α0=sup0≤t≤+∞α(t)<0. \left\{ {\matrix{ {{C_\mu } = v + (1 - v)\exp ({\alpha _0}\mu ) \in (0,1),} \hfill \cr {v = \mathop {\sup }\limits_{0 \le t \le + \infty } (\beta (t)/\left| {\alpha (t)} \right|) < 1,{\alpha _0} = \mathop {\sup }\limits_{0 \le t \le + \infty } \alpha (t) < 0.} \hfill \cr } } \right.

These results represent the stability, generalised shrinkage and asymptotic stability of solutions of rigid functional differential equations. This work provides a unified theoretical basis for the stability analysis of solutions of rigid ordinary differential equations, rigid delay differential equations, rigid integral differential equations and other functional differential equations encountered in practical problems and is also the theoretical basis for the study of numerical methods of rigid functional differential equations [11].

As one of the special cases of the above general theory, for a nonlinear rigid delay differential equation with a single delay τ(t) in a finite dimensional Euclidean space C^m, in 1989, Torelli showed that the stability inequality (3) holds for sup0≤t≤+∞(α(t)+β(t))≤0 \mathop {\sup }\limits_{0 \le t \le + \infty } (\alpha (t) + \beta (t)) \le 0 . Zennaro previously obtained asymptotic stability results in 1997 (5). However, in addition to all the assumptions in the corresponding theorem mentioned above, their results also require the following more stringent additional assumptions.

There exists a constant τ₀ > 0 such that τ(t)≥τ0 ∀t∈[0,+∞); \tau (t) \ge {\tau _0}\;\;\;\;\;\;\;\;\;\;\;\;\;\forall t \in \left[ {0, + \infty } \right); t − τ(t) is a strictly increasing function on the interval [0,+∞).

Thus it can be seen that even for the well-studied differential equations at home and abroad, the results directly derived from the above general theory are more general and profound than the existing results.

This section recommends several classes of numerical methods that can be used to solve rigid Volterra functional differential equations with good performance in all aspects.

The initial boundary value problem of one-dimensional parabolic differential equations with a delay term is considered (10) {∂u∂t=∂2u(x,t)∂x2=2e−t−et/2u(x,t2)+∫0tu(x,τ)dτ, 0<x<1,0<t≤1,u(0,t)=u(1,t)=0, 0≤t≤1,u(x,0)=x−x2, 0<x<1, \left\{ {\matrix{ {{{\partial u} \over {\partial t}} = {{{\partial ^2}u(x,t)} \over {\partial {x^2}}} = 2{e^{ - t}} - {e^{t/2}}u(x,{t \over 2}) + \int_0^t u(x,\tau )d\tau ,} \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{0 < x < 1,0 < t \le 1,} \hfill \cr {u(0,t) = u(1,t) = 0,\;\;\;\;\;\;\;\;\;\;\;0 \le t \le 1,} \hfill \cr {u(x,0) = x - {x^2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,0 < x < 1,} \hfill \cr } } \right. The problem has a unique true solution to u(x, t) = (x − x²)e^−t. Therefore, we can replace the second-order spatial derivative with the second-order central difference quotient in the network domain x_i = i/N, i = 1(1)N − 1; there is no local truncation error, where N is any given natural number. Denote u_i(t) = u(x_i, t), Δx = 1/N, thus obtaining the semi-discrete approximation of problem (10) (11) {dui(t)dt=1Δx2(ui−1(t)−2ui(t)+ui+1(t))+2exp(−t) −exp(t2)ui(t2)+∫0iui(τ)dτ,u0(t)≡uN(t)≡0,ui(0)=iΔx(1−iΔx),i=1,2,⋯,N−1, \left\{ {\matrix{ {{{d{u_i}(t)} \over {dt}} = {1 \over {\Delta {x^2}}}({u_{i - 1}}(t) - 2{u_i}(t) + {u_{i + 1}}(t)) + 2\exp ( - t)} \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ - \exp ({t \over 2}){u_i}({t \over 2}) + \int_0^i {u_i}(\tau )d\tau ,} \hfill \cr {{u_0}(t) \equiv {u_N}(t) \equiv 0,} \hfill \cr {{u_i}(0) = i\Delta x(1 - i\Delta x),i = 1,2, \cdots ,N - 1,} \hfill \cr } } \right. Make f(t,u,ψ(⋅))=[f1(t,u,ψ(⋅)),f2(t,u,ψ(⋅)),⋯,fN−1(t,u,ψ(⋅))]T, f(t,u,\psi ( \cdot )) = {\left[ {{f_1}(t,u,\psi ( \cdot )),{f_2}(t,u,\psi ( \cdot )), \cdots ,{f_{N - 1}}(t,u,\psi ( \cdot ))} \right]^T}, Among them fi(t,u,ψ)=12Δx2(ui+1−2ui+ui−1)+2e−t−et/2ψi(t2)+∫0tψi(τ)dτ,ψ=[ψ1,ψ2,⋯,ψN−1]T∈CRN−1[0,1],u0=uN=0. \matrix{ {{f_i}(t,u,\psi ) = {1 \over {2\Delta {x^2}}}({u_{i + 1}} - 2{u_i} + {u_{i - 1}}) + 2{e^{ - t}} - {e^{t/2}}{\psi _i}({t \over 2}) + \int_0^t {\psi _i}(\tau )d\tau ,} \cr {\psi = {{\left[ {{\psi _1},{\psi _2}, \cdots ,{\psi _{N - 1}}} \right]}^T} \in {C_{{R^{N - 1}}}}\left[ {0,1} \right],{u_0} = {u_N} = 0.} \cr } Then problem (11) can be written in the form of initial value problem (1) of Volterra functional differential equation in space R^m; according to the standard inner product, it satisfies the condition (2)’; here, m = N −1, τ = 0, T = 1, α=−4N2sin2π2N≅−π2 \alpha = - 4{N^2}\mathop {\sin }\nolimits^2 {\pi \over {2N}} \cong - {\pi ^2} (when N is large), β=1+e \beta = 1 + \sqrt e (here e is the base of the natural log). Therefore, the B theory of the numerical method of Volterra functional differential equations introduced in Section 2 of this paper can be directly applied to this special case, and various efficient numerical methods recommended in Section 3 of this paper can be as well used to solve the problem (11) [12].

As an example, we consider the second-order and third-order real eigenvalue implicit multistep Runge-Kutta method (RMRK3) for solving rigid ordinary differential equations proposed in the literature. (12) {Y=hC11F(Y)+C12y(n),yn+1=hγTF˜(Y)+αTy(n), \left\{ {\matrix{ {Y = h{C_{11}}F(Y) + {C_{12}}{y^{(n)}},} \hfill \cr {{y_{n + 1}} = h{\gamma ^T}\tilde F(Y) + {\alpha ^T}{y^{(n)}},} \hfill \cr } } \right. Here {y(n)=[yn−2T,yn−1T,ynT]T≈[y(tn−2)T,y(tn−1)T,y(tn)T]T,Y=[y1T,y2T]T≈[y(tn+μ1h)T,y(tn+μ2h)T]T,F˜(Y)=[f(tn+μ1h,Y1)T,f(tn+μ2h,Y2)T]T,C11=[cij11]∈R2×2,C12=[cij12]∈R2×3,γT=[γ1,γ2],αT=[α1,α2,α3], \left\{ {\matrix{ {{y^{(n)}} = {{\left[ {y_{n - 2}^T,y_{n - 1}^T,y_n^T} \right]}^T} \approx {{\left[ {y{{({t_{n - 2}})}^T},y{{({t_{n - 1}})}^T},y{{({t_n})}^T}} \right]}^T},} \hfill \cr {Y = {{\left[ {y_1^T,y_2^T} \right]}^T} \approx {{\left[ {y{{({t_n} + {\mu _1}h)}^T},y{{({t_n} + {\mu _2}h)}^T}} \right]}^T},} \hfill \cr {\tilde F(Y) = {{\left[ {f{{({t_n} + {\mu _1}h,{Y_1})}^T},f{{({t_n} + {\mu _2}h,{Y_2})}^T}} \right]}^T},} \hfill \cr {{C_{11}} = \left[ {c_{ij}^{11}} \right] \in {R^{2 \times 2}},{C_{12}} = \left[ {c_{ij}^{12}} \right] \in {R^{2 \times 3}},} \hfill \cr {{\gamma ^T} = \left[ {{\gamma _1},{\gamma _2}} \right],{\alpha ^T} = \left[ {{\alpha _1},{\alpha _2},{\alpha _3}} \right],} \hfill \cr } } \right. Among them {c1111≈0.915809216730669,c1211≈−0.453336810549659,c2111≈0.0149519583665559,c2211≈0.741962909796344,c1112≈0.193708609271523,c1212≈0,c1312≈0.806291390728476,c2112≈−0.121542565918549,c2212≈0,c2312≈1.12154256591855,γ1≈0.653011933174220,γ2≈0.398988066825780,α1=0.026,α2=0,α3=0.974,μ1≈0.075055187637962,μ2=1. \left\{ {\matrix{ {c_{11}^{11} \approx 0.915809216730669,c_{12}^{11} \approx - 0.453336810549659,} \hfill \cr {c_{21}^{11} \approx 0.0149519583665559,c_{22}^{11} \approx 0.741962909796344,} \hfill \cr {c_{11}^{12} \approx 0.193708609271523,c_{12}^{12} \approx 0,c_{13}^{12} \approx 0.806291390728476,} \hfill \cr {c_{21}^{12} \approx - 0.121542565918549,c_{22}^{12} \approx 0,c_{23}^{12} \approx 1.12154256591855,} \hfill \cr {{\gamma _1} \approx 0.653011933174220,{\gamma _2} \approx 0.398988066825780,{\alpha _1} = 0.026,} \hfill \cr {{\alpha _2} = 0,{\alpha _3} = 0.974,{\mu _1} \approx 0.075055187637962,{\mu _2} = 1.} \hfill \cr } } \right. These coefficients are calculated according to the techniques provided and the relevant formulae in the literature. In these two papers, it is proved that the RMRK3 method is algebraically stable, diagonally stable, with order 2 and weak order 3. It is easy to know from B theory that the numerical method of Volterra functional differential equation obtained by assembling piecewise quadratic polynomial interpolation operator is B-stable, B compatible and has the best B convergence order 2 and B convergence order 3. For convenience, we still denote the obtained method as RMRK3. In the actual calculation of each integral step, the linear equations caused by the simplified Newton iteration can be decomposed into two independent low-order linear equations by Butcher transform for solving, which can be either serially calculated or parallel calculated by two processors to improve the computational efficiency.

In addition, for comparison, we also consider Volterra's Dirk3 method for functional differential equations, which is derived from the well-known 2nd and 3rd order single-step diagonal implicit Runge-Kutta method for solving rigid ordinary differential equations (13) {Y1=yn+hγf(tn+γh,Y1)Y2=yn+h[(1−2γ)f(tn+γh,Y1)+γf(tn+(1−γ)h,Y2)]yn+1=yn+h2[f(tn+γh,Y1)+f(tn+(1−γ)h,Y2)] \left\{ {\matrix{ {{Y_1} = {y_n} + h\gamma f({t_n} + \gamma h,{Y_1})} \cr {{Y_2} = {y_n} + h[(1 - 2\gamma )f({t_n} + \gamma h,{Y_1}) + \gamma f({t_n} + (1 - \gamma )h,{Y_2})]} \cr {{y_{n + 1}} = {y_n} + {h \over 2}[f({t_n} + \gamma h,{Y_1}) + f({t_n} + (1 - \gamma )h,{Y_2})]} \cr {} \cr } } \right. Here γ=(3+3) \gamma = \left( {3 + \sqrt 3 } \right) is assembled by piecewise quadratic polynomial interpolation operator. Note that the Dirk3 method for functional differential equations is neither B-stable nor B-convergent, since its parent method (13) is only A-stable and classically convergent to the third-order.

In order to make a fair and reasonable comparison, the two methods are used to solve the problem (11) (N=5000) on the same Dell Optiplex GX270 computer with the same fixed step size h, and both are calculated in a serial way, and the simplified Newton iteration is used to solve the nonlinear equations. The direct method is used to solve the linear equations, and the same iteration termination strategy is adopted. The maximum overall error E and the calculation time T of the obtained numerical results can be seen in Table 1. It can be clearly seen that the computational efficiency of the B-stabilised method RMRK3 is indeed higher than that of the DIRK3 method.

In this paper, based on the rigid normal differential equation and rigid general functional equation of financial accounting measurement research, effectively solving the numerical solution of the two equations and the solution of the equation are expressed; then the reliability and practicability of the two equations in financial accounting measurement are expressed.