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Respected Dear Academicians,

May I request you to kindly provide me an opportunity for joint research work on Geomathematics Seismic wave propagation modeling and simulation , hyperbolic partial differential equations solution, high performance computing

ExaHyPe Hyperbolic PDE engine

With best regards

Sunjay

The study of fractional hyperbolic partial differential equations (PDEs) is a relatively new and active area of research in mathematics and physics. In particular, fractional hyperbolic PDEs have been studied extensively in recent years. These equations involve fractional derivatives in the time variable, which lead to memory effects and long-range interactions. They have been used to model a wide range of physical phenomena, such as wave propagation in porous media, viscoelasticity, and anomalous diffusion.

The study of fractional hyperbolic PDEs is still an active area of research, with ongoing work focused on developing new numerical methods for solving these equations, analyzing the existence and uniqueness of solutions, and exploring the physical implications of these models. There is also significant interest in developing applications of these models in various fields, such as geophysics, acoustics, and biomedical engineering. Overall, the study of fractional hyperbolic PDEs is a rapidly evolving field with many exciting developments and potential applications.

Here it is our hope that we can discuss the associated breakthroughs as well as challenging aspects of the Fractional hyperbolic PDE.

What type of numercial methods are there to solve PDE of the sorts of:

$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$

$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t)=G_2(t) , \frac{\partial u(0,t)}{\partial x} = H_2(t)$$

Where $f,g,F,u \in C^\infty(x,t) , \ G_i,H_i \in C^\infty$.

Specifically I had in mind the PDE:

$$u_{xx}u^3-\sin(xt)u_{tt} = u$$

But the general PDE is as above; I looked at Polyanin's second edition Handbook of Nonlinear PDE table of conetents, and didn't find something similar, obviously I look at numerical solutions since an analytical solution doesn't seem plausible, but if there is I wouldn't mind.

In the context of scalar hyperbolic partial differential equations (PDEs), the concept of "stiffness" generally refers to the behavior of the solution as the time step or grid spacing is decreased. Specifically, a PDE is said to be "stiff" if the solution changes rapidly over small time scales or distances, requiring a correspondingly small time step or grid spacing in order to accurately capture the behavior of the solution.

Stiffness is a particularly important concept in the numerical solution of hyperbolic PDEs, as these equations often exhibit sharp gradients or shocks that can be difficult to accurately capture using standard numerical methods. Stiffness can lead to numerical instability, slow convergence, and other issues that can make it difficult to obtain accurate solutions to hyperbolic PDEs. As such, developing numerical methods that can handle stiffness is an important area of research in computational mathematics and scientific computing.

I need to know this answer is correct? how to calculate is?

Can you suggest any reference please

I need to solve the Fokker-Planck equation numerically. I need to know which method is the best method for solving this kind of PDE? among Finite difference, Finite element, Finite volume, Spectral methods, or any other kinds of methods?which ones is the best choice? and by the "best Method" I mean stable and fast numerical method.

Since explicit methods face stability issues, why is it that implicit methods are rarely used for time dependent hyperbolic problems?

The nonphysical measure-valued solution of hyperbolic PDE can not be reached by any viscous solution, which can only dissipative energy but not creates.

I have a model of sintering, the results of which have a hyperbolic PDE(conservation of mass), and two parabolic equations (Navier's equation and one for temperature). Normally I would try and solve these equations using pdepe, but pdepe doesn't deal with hyperbolic equations(like the conservation of mass).

I tried a naive approach by using central differences for my spacial differences, but this just made everything unstable. As the conservation of mass is hyperbolic, I have to use a different way of descritising the spatial derivative. Is it just this simple? The centred difference for the first order derivative is accurate to O(dx^2), if I reduce the order to a simple downwind(or upwind) for the spatial derivative in the hyperbolic equation, will that make the system unstable, or do I have to so this splitting method stuff?

In the realm of solving hyperbolic partial differential equations (PDEs) through physics-informed Neural Networks (PINNs), my knowledge extends up to the most recent update. To my knowledge, the approach to addressing hyperbolic PDEs with PINNs involves the integration of governing equations, initial conditions, and boundary conditions into the loss function. This loss function plays a pivotal role in guiding the training process of the neural network, enabling it to approximate solutions to the PDE.

Notably, it's essential to consider various facets of neural network error, including optimization error, approximation error, and estimation error. The approximation error denotes the difference between the exact functional map and the neural network's mapping function on a given network architecture. Estimation error, on the other hand, emerges when the network is trained on a finite dataset to establish a mapping for the target domain. The generalization error encompasses both approximation and estimation errors, defining the accuracy of the neural network's predicted solution based on the provided dataset.

In the current landscape, considerable research efforts are dedicated to optimizing the coefficients associated with error terms. Additionally, there has been a development known as Conservative PINN (CPINN). However, for someone seeking to delve into this domain, it's understandable to feel confused about where to start. Seeking guidance from the community or experts in the field could be immensely beneficial in navigating the complexities and determining an effective starting point.

Fully developed turbulence exhibits long-range mean square velocity correlations <(v||(L)-v||(0))^2>=K*L^(-3/2) (the two-thirds law). This is consistent with the 3d-directed percolation model.

However, is this consistent with an elliptical PDE that is supposed to describe subsonic flows?