How do you create a partial differential equation in Word?
Answers from top 14 papers
More filters
| Papers (14) | Insight |
|---|---|
| The MOL system of ordinary differential equations is shown to be a valid representation of the partial differential equation. | |
14 Citations | It is shown that the problem generally requires the solution of a partial differential equation. |
| It is shown that such a model is equivalent to an advection-diffusion partial differential equation model. | |
24 Citations | We show that the model follows a nonlinear parabolic type partial differential equation. |
46 Citations | The method, as applied to a partial differential equation with three independent variables, reduces the equation to an overdetermined system of ordinary differential equations having explicit solutions, and a joining system of partial differential equations with two independent variables which can easily be solved. |
| The proposed technique is an efficient and powerful mathematical method for solving a wide range of nonlinear partial differential equation. | |
03 Jun 2020 | Such an approach gives more versatility in comparison with other commonly used methods of data-driven partial differential derivation by making fewer restrictions on the resulting equation. |
| This new method presents a wider applicability for handling the nonlinear partial differential equations. | |
31 Citations | The used methods present a wider applicability for handling the nonlinear partial differential equations. |
| We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. | |
16 Citations | The method can also solve other nonlinear partial differential equations. |
| The method is straightforwardly applicable to various types of partial differential equations. | |
22 Citations | The method can also be applied to other nonlinear partial differential equations. |
21 Citations | The method works well for ordinary and partial differential equations. |
Related Questions
What is differential equation?5 answersA differential equation is an equation that involves an unknown function and its derivatives with respect to one or more independent variables. If the equation involves only one independent variable, it is termed an ordinary differential equation; if it involves more than one independent variable, it is called a partial differential equation. Differential equations are fundamental in modeling systems across various fields like engineering, physics, biology, and economics. While finding exact solutions for these equations is generally challenging, numerical solutions play a crucial role in practical applications, aiding in the design and understanding of complex systems. The study of differential equations is vast and significant, offering a powerful tool for quantitatively explaining diverse physical and abstract systems.
What is the role of partial differential equations in chemistry?5 answersPartial differential equations (PDEs) play a crucial role in modeling various phenomena in chemistry. They form the basis for describing the change of velocity, temperature, and concentration in systems with multiple independent variables. In chemical engineering, PDEs are commonly encountered, especially first and second-order equations, which are essential for understanding transport phenomena. Numerical methods like finite difference and finite element analysis are employed to solve these PDEs when analytical solutions are challenging to obtain. Additionally, PDEs are used to model diffusive-advective transport equations in chemical systems, with applications in solving steady-state conditions and addressing challenges like hyperbolic PDE systems and discontinuous solutions. Overall, PDEs are fundamental tools in chemistry for predicting and understanding complex dynamics at various spatial and temporal scales.
How can Floquet theory be used to solve partial differential equations?5 answersFloquet theory can be used to solve partial differential equations by applying the concept of Floquet transition matrix (FTM). The FTM is obtained by considering one polynomial basis function at a time as the history function and stacking the coefficients of the corresponding solutions of the differential equation. The largest magnitude eigenvalue of the FTM determines the stability of the system. This approach has been successfully applied to study the stability of time-periodic delay differential equations (DDEs). Additionally, Floquet theory has been used to study the effects of laser light coupling to an electron-hole pair confined in a quantum wire, where the renormalization of binding energy leads to distinct features in the Floquet energy spectra. Furthermore, a fractional generalization of the Floquet theorem has been proposed for fractional Schrödinger equations, allowing for reduction to standard quantum mechanics with time-dependent Hamiltonians. A novel method using Floquet theorem and a time-dependent unitary transformation has also been developed to obtain exact solutions of the Schrödinger equation for periodically driven systems. Finally, Floquet theory has been applied to quaternion-valued differential equations, providing a Floquet normal form for linear QDEs with periodic coefficients and studying the stability of quaternionic periodic systems.
How can graph neural networks be used to fit partial differential equations?5 answersGraph neural networks (GNNs) can be used to fit partial differential equations (PDEs) by leveraging their ability to accurately represent irregularly meshed objects and learn their dynamics. GNNs have shown promise in solving physics problems by enhancing neural networks with knowledge of physical equations. They can be used as fast and accurate alternatives for principled solvers when applied to standard equations with regular solutions. GNNs can also be used as an interactive PDE solver, adjusting boundary conditions in real-time and remaining close to the baseline solution. By representing meshes naturally as graphs and formulating a physics-based loss, GNNs provide an unsupervised learning framework for PDEs. Additionally, GNNs can learn mappings between input data to PDEs and their solutions, generalizing among different approximation methods and levels of resolution and discretization.
What are some examples of differential equations in real life in engineering?5 answersDifferential equations are widely used in engineering to model various real-life phenomena. For example, in the field of feedback control and fault detection, real-time tracking filtering and differentiation estimation of signals are commonly performed using differential equations. Differential equations are also used in physics and engineering to describe exponential growth and decay, population growth, and change in investment return over time. In chemical and food technology, differential equations are used to solve various problems, such as reaction rates and concentration of substances. Additionally, differential equations are used in formulating the fundamental laws of nature and solving technological problems. Therefore, differential equations play a crucial role in engineering by providing mathematical models for understanding and solving real-life problems.
Which mathematician made contributions to partial differential equations and mechanics and was the first woman to obtain a doctorate in maths?6 answers