CLASSIFICATION OF FUNCTIONS IN ASSOCIATION WITH DIFFERENTIAL EQUATIONS

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples.

Non-homogeneous differential equations play a crucial role in modelling real-world phenomena across physics, engineering, and economics. Two classical analytical methods for solving such equations are the Method of Undetermined Coefficients and the Method of Variation of Parameters. This paper explores both techniques in detail, highlighting their theoretical foundations, computational procedures, and practical applications. While undetermined coefficients provide a straightforward approach for equations with specific forcing functions such as polynomials, exponentials, and trigonometric terms, variation of parameters offers a more general framework applicable to a wider class of functions. Comparative analysis demonstrates that undermined coefficients are efficient but limited in scope, whereas variation of parameters, though computationally intensive, ensures universality. Case studies illustrate the strengths and limitations of each method, providing insights into their relevance in applied mathematics and engineering contexts. The study concludes by emphasising the complementary nature of Undetermined Coefficients and Variation of Parameters, suggesting that a combined understanding of these methods is essential for advanced problem-solving in differential equations.

Annealed cooperative–competitive learning of Mahalanobis-NRBF neural modules for nonlinear and chaotic differential function approximation

Higher-order linear differential equations have great significance theoretically and practically. This paper solves a third-order linear differential equation with constant coefficients by method of undetermined coefficients. Its strengths and limitations were analysed. Analysis suggests that the method of undetermined coefficients requires that the inhomogeneous term be algebraic, exponential, or sinusoidal functions only. Additionally, if the inhomogeneous term is a solution of the differential equation itself, may yield a linear combination of . In such case, the prediction for will be modified by multiplying the variable however many times necessary.

In this research article we have used the method of undetermined coefficients with modification rule and method of variation of parameters to solve linear ordinary differential equations and found that the method of variation of parameters is the easiest method to use.

Solving function approximation problem is to appropriately find the relationship between dependent variable and independent variable(s). Function approximation algorithms normally require sufficient amount of samples to approximate a function. However, insufficient samples may result in unsatisfactory prediction to any function approximation algorithms. It is due to the failure of the function approximation algorithms to fill the information gap between the available and very limited samples. In this study, a function approximation algorithm which is based on Nadaraya-Watson Kernel Regression (NWKR) is proposed for approximating a non-linear function with small samples. Gaussian function is chosen as a kernel function for this study. The results show that the NWKR is effective in the case where the target function is non-linear and the given training sample is small. The performance of the NWKR is compared with other existing function approximation algorithms, such as artificial neural network.

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

Over the years, the teaching of differential equations in the engineering programs at the University of Antofagasta, Chile, has faced various challenges, particularly regarding the learning of the method of undetermined coefficients. This method, used to find particular solutions in third-order linear differential equations, has been identified as one of the most complex topics for students. The difficulty lies primarily in determining the general solution of these equations, which requires a high level of understanding and advanced mathematical skills. To address these challenges and improve the learning process, innovative technological tools such as GeoGebra applets have been developed. These applications allow students to interact dynamically with the concepts and procedures of the method of undetermined coefficients, facilitating their understanding and application. The results obtained after implementing these tools have been highly positive, showing a significant improvement in students' learning and motivation. Additionally, this work incorporates the use of artificial intelligence, specifically through DeepSeek. This AI has proven to be an effective tool for assisting in the determination of particular solutions, which not only simplifies the resolution process but also fosters students' interest and active participation in their tasks and studies. The integration of these technologies not only strengthens learning but also modernizes the teaching of differential equations, aligning it with the current demands of engineering education.

In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination; and these systems suffice for linear differential equations. But when we enter upon the consideration of nonlinear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the following memoir. Professor Boole has given, in his ‘Treatiseon Differential Equations/ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole’s treatise, is equivalent to finding the conditions that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for those conditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non linear functions by linear functions of the symbol of differentiation.

A unified framework is presented for nonlinear (and, in particular, linear) system and signal analysis, whereby a number of problems involving approximation and inversion of nonlinear functions, nonlinear functionals, and nonlinear operators, are cast in a reproducing kernel Hilbert space (RKHS) setting, and solved by orthogonal projection methods. The RKHS's used for the above purpose are the "arbitrarily weighted Fock spaces" introduced by de Figueiredo and Dwyer in II] and called in the present paper "generalized Fock (GF) spaces". These spaces consist of polynomials or power series in one or more scalar variables, or of finite (polynomic case) or infinite Volterra functional series in one or more functions, or of finite or infinite Volterra operator series in one or more functions. In each case, the space is equipped with an appropriate weighted inner product, with the option of making the choice of weights depend on the particular problem under consideration. These developments are illustrated by means of various applications, in particular, the modeling of semiconductor device characteristics, best approximation of nonlinear systems, and cancellation of large nonlinear distortion in signals propagating through electronic equipment.

Considering nonlinear control, identification of nonlinearities has become essential. Thus, a universal function approximator has to be found which is able to approximate an unknown static1 nonlinear function. For this objective, various neural network and also fuzzy approaches are offered.

In this paper, we propose an approach for constructing conservative and maximum-principle-preserving finite volume schemes by using the method of undetermined coefficients, which depend nonlinearly on the linear non-conservative one-sided fluxes. In order to facilitate the derivation of expressions of these undetermined coefficients, we explicitly provide a simple constriction condition with a scaling parameter. Such constriction conditions can ensure the final schemes are exact for linear solution problems and may induce various schemes by choosing different values for the parameter. In particular, when this parameter is taken to be 0, the nonlinear terms in our scheme degenerate to a harmonic average combination of the discrete linear fluxes, which has often been used in a variety of maximum-principle-preserving finite volume schemes. Thus our method of determining the coefficients of the nonlinear terms is more general. In addition, we prove the convergence of the proposed schemes by using a compactness technique. Numerical results demonstrate that our schemes can preserve the conservation property, satisfy the discrete maximum principle, possess a second-order accuracy, be exact for linear solution problems, and be available for anisotropic problems on distorted meshes.

Determination of asset prices with an investment-specific technology model: Implications for the equity premium puzzle

Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.

Analytical solution for the 1D consolidation of unsaturated multi-layered soil