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Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly…
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $\Delta^{\frac{\alpha}{2}} u=\rho(x)\,\varphi(u)$ admits nonnegative nontrivial bounded solutions in the whole space…
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical…
Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…
We investigate here all the possible invariant metric functions under the action of various kinds of semi-direct product Poincar\'e subgroups and their deformed partners. The investigation exhausts the possible theoretical frameworks for…
In our previous work, we have defined a nonlinear connection of Finsler manifold which preserves the Finsler metric $L=L(x,dx)$. To make the method easier and more useful in applications, moving frame (vielbein) $\theta^a={e^a}_\mu dx^\mu$…
We develop a new method of solving Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is…
The main goal of this article is to study a Calder\'on type inverse problem for certain viscous nonlocal wave equations. We show that the partial Dirichlet to Neumann map uniquely determines on the one hand linear perturbations and on the…
We address the study of nonlocal gradients defined through general radial kernels $\rho$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function.…
This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and…
A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a…
The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the…
A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincar\'e's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this…
For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The…
The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally…
We study a particular system of partial differential equations in which the harmonic, the divergence and the gradient operators of the unknown functions appear (harmonic-divgrad system). Using the Killing Hopf theorem and leveraging the…
We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be…
We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…