In order to solve the problems of financial accounting measurement quickly and accurately, this paper starts the analysis from the perspective of mathematics and finance and establishes the differential equation and the generalised functional equation for the related numerical analysis through mathematical knowledge. The results show that the limit and integral of rigid differential equation and the rigid generalised functional equation can improve their role and status in the financial accounting measurement environment so that they can be more widely used in the financial accounting measurement environment and promote the development of the financial accounting environment.

#### Keywords

- rigid differential equation
- rigid general functional subequation
- numerical analysis
- the financial accounting

#### MSC 2010

- 97B10

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’ =

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 (

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 (

Let X be a real or complex Banach space and ‖•‖ denote the norm therein. For any closed interval, the symbol _{X}_{X}_{X}_{1}(_{2}(_{1}(_{2}(

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

That's true, right there

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

_{k}_{0} = 0 is such that

Here,

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 ^{m}

There exists a constant _{0} > 0 such that

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.

Both the backward Euler and trapezoid methods are bootstrapping and can be used to provide additional starting values for the following numerical methods.

The k order parallel multi-valued mixed method for rigid ordinary differential equation is defined as

They are B-stable, B-compatible, B-convergent and extremely stable at ∞. Parallel computations for large-scale rigid problems have roughly the same computational speed as the backward Euler method, but additional starting values are needed. Because of its excellent stability, it is suitable for the high-precision calculation of nonlinear rigid problems.

Since the calculation speed of single-step fully hidden Runge-Kutta method above level 2 is relatively slow, for example, the number of computations for each integration step of the second-order and third-order one-step Radau2A Runge-Kutta method under Butcher transform is at least four or five times as much as that of the backward Euler method, respectively, and so we do not recommend the use of such methods. We also do not recommend the single-step fully implicit continuous Runge-Kutta methods. Although they are in practice from the beginning, they not only have the disadvantages of the single-step fully implicit Runge-Kutta methods mentioned above but also cause order reduction due to interpolation with internal level values.

The initial boundary value problem of one-dimensional parabolic differential equations with a delay term is considered
^{2})^{−t}. Therefore, we can replace the second-order spatial derivative with the second-order central difference quotient in the network domain _{i}_{i}_{i}^{m}

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.

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

In order to make a fair and reasonable comparison, the two methods are used to solve the problem (11) (

Integral step size h, corresponding maximum overall error e and calculation time t

1/200 | 1.70e-8 | 7.20 | 2.57e-7 | 6.8 |

1/400 | 2.18e-9 | 14.5 | 4.26e-8 | 13.7 |

1/600 | 6.52e-10 | 21.7 | 1.48e-8 | 20.5 |

1/800 | 2.77e-10 | 29.8 | 6.93e-9 | 27.4 |

1/1000 | 1.42e-10 | 37.9 | 3.84e-9 | 34.2 |

RMRK3, real eigenvalue implicit multistep Runge-Kutta method.

Note that for large-scale rigid problems, the RMRK3 method can be computed in parallel with both processors, which is close to the backward Euler method and faster than any other commonly used multistage methods, such as the DIRK3 method, which is significantly faster because it can be computed only serially.

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.

#### Integral step size h, corresponding maximum overall error e and calculation time t

1/200 | 1.70e-8 | 7.20 | 2.57e-7 | 6.8 |

1/400 | 2.18e-9 | 14.5 | 4.26e-8 | 13.7 |

1/600 | 6.52e-10 | 21.7 | 1.48e-8 | 20.5 |

1/800 | 2.77e-10 | 29.8 | 6.93e-9 | 27.4 |

1/1000 | 1.42e-10 | 37.9 | 3.84e-9 | 34.2 |

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