A Maximum-Principle-Preserving Finite Volume Scheme for Diffusion Problems on Distorted Meshes

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.

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.

Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards’ equation on unstructured meshes

We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.

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.

A novel cell-centered finite volume scheme with positivity-preserving property for the anisotropic diffusion problems on general polyhedral 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

Design of transient thermo-mechanical systems is a challenging task often encountered during the design of high precision machines and instrumentation. Topology optimization can provide valuable insight during the design process, however, for large scale problems the backward time integration required to obtain adjoint sensitivity information is undesired. Previous work has illustrated how the introduction of a reduced modal basis allows to eliminate the backward time integration to obtain the adjoint variables. In order to reduce computational effort further, additional reduction approaches are considered. The focus is specifically on design cases where the relevant heat loads can be expressed or approximated analytically by combinations of harmonic, polynomial or exponential functions of time. Using the method of undetermined coefficients, an exact particular solution is obtained using the full system. Then, the corresponding homogenous solution is expressed using a reduced modal basis, for which a relatively small set of modes is required to obtain an accurate approximation. For the cases where the time component of the heat loads are expressed by the considered analytical functions, the backward time integration is eliminated from the calculation of the design sensitivities, while the forward integration is handled by convolutions.

In this study, two sixth-order compact finite difference schemes have been considered for solving the Burgers equation. The main difference of these schemes lies in the calculation of second-order derivative terms, which is obtained by applying the first-order operator twice and the method of undetermined coefficients. The aim is to comparison these schemes in terms of computational accuracy for solving the Burgers equation with difference viscosity values, especially for very small viscosity values. The results show that both schemes achieve almost the same accuracy for large viscosity values and second method is more accurate for moderate viscosity values, but both schemes are failed for very small viscosity values. However, when both schemes coupled low-pass filter for very small viscosity values, both schemes can well inhibit the problem.

The Method of Undetermined Coefficients

Purpose. To improve the efficiency of fine particle classification in vibrating screens by determining the rational dynamic parameters of designs with spatial oscillations of the working body. Methods. A dynamic model of the vibrating screen was developed on the basis of the Lagrange equations of the second kind, taking into account mass, inertial, and elastic-damping characteristics. The amplitude–frequency characteristics were determined by the method of undetermined coefficients, while non-elastic resistances were considered according to Volterra’s principle. A parametric analysis of the influence of the screen geometry and exciter position was carried out using mathematical modelling and analytical relationships. The adequacy of the model was verified by comparison with known experimental data. Findings. For the first time, the influence of the mutual arrangement of the drive and the dimensions of the screen on the ratio of linear and angular displacements in systems with spatial oscillations has been substantiated. This makes it possible to purposefully form particle motion trajectories and optimise their transport velocity. It was shown that correct adjustment of the parameters reduces the risk of local overloads, ensures uniform loading of the screening surface, and increases the operational stability of the equipment. Originality. Dependencies of vibration amplitudes on the drive installation location and geometric parameters have been obtained. Regularities in the variation of the vibration displacement coefficient across the width of the screening surface have been determined. The proposed approach makes it possible to predict the behaviour of the working body under variable loading conditions and to formulate recommendations aimed at improving the efficiency of separation processes. Practical value. The developed recommendations can be used in the design of vibrating screens for the mining and mineral processing industry. Their application will contribute to increased productivity and selectivity, reduced energy consumption of processes, lower maintenance costs, and enhanced equipment reliability. The obtained conclusions are also useful for other industries where vibration technologies are applied, in particular for the processing of bulk construction materials and in the food industry.

In this work, the multivariate analogue to the univariate Wold’s theorem for a purely non-deterministic stable vector time series process was presented and justified using the method of undetermined coefficients. By this method, a finite vector autoregressive process of order [] was represented as an infinite vector moving average () process which was found to be the same as the Wold’s representation. Thus, obtaining the properties of a process is equivalent to obtaining the properties of an infinite process. The proof of the unbiasedness of forecasts followed immediately based on the fact that a stable VAR process can be represented as an infinite VEMA process.