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We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs…
The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that that the numbers of symmetries and Lie brackets are reduced significantly as compared to the nonfractional order…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…
Systems of nonlinear ordinary differential equations are constructed, for which the general solution is algebraically expressed in terms of a finite number of particular solutions. Expressions of that type are called the nonlinear…
A family of parabolic Szekeres-Szafron class II solutions in local thermal equilibrium is studied and their associated thermodynamics are obtained. The subfamily with the hydrodynamic behavior of a generic ideal gas (defined by the equation…
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental…
In this paper we consider the overdetermined Cauchy problem for the heat equation. We prove that if the problem has a nontrivial nonnegative solution with a certain sequence of similar level sets, then the solution must be radially…
We first study the so-called Heat equation with two families of elliptic operators whichare fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equationwith operators including the "large" eigenvalues has strong…
We present an efficient way to compute diagonal and off-diagonal n-point correlation functions for quantum spin-systems within the loop algorithm. We show that the general rules for the evaluation of these correlation functions take an…
New problem is studied that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. Method is discussed to construct nonlinear ordinary differential equations with exact solutions. Main…
The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side…
Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the…
We consider non-linear time-fractional stochastic heat type equation $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$,…
This work obtains a fixed-point equation for the solution of linear parabolic partial differential problems based on solutions to heat problems. This is a pointwise equality, so we have required non-standard techniques that involve the…
We study anomalous transport in a one-dimensional system with two conserved quantities in presence of thermal baths. In this system we derive exact expressions of the temperature profile and the two point correlations in steady state as…
Using the physical background based on the earlier author's researches the new method of the ODE and PDE integration is proposed. As illustrations of the method the general solution for free oscillator and for heat equation are derived. As…
An exact linear response expression is obtained for the heat current in a classical Hamiltonian system coupled to heat baths with time-dependent temperatures. The expression is equally valid at zero and finite frequencies. We present…
We prove two stability results for the scale invariant solutions of the nonlinear heat equation $\partial_t u=\Delta u - |u|^{p-1}u$ with $1<p<1+{2\over n}$, $n$ being the spatial dimension. The first result is that a small perturbation of…
Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the…
The role of the algebraic method has long been understood in shedding light on the topological structure of sets. However, when the set is a simplicial complex and host to a dynamical process, in particular the trajectory of a canonically…