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Let u be a function of several real variables (t, x 1, x 2, …, x n ): $$ \begin{gathered} u:\mathbb{R}^ + \times \mathbb{R}^n \quad \,\, \to \mathbb{R} \hfill \\ \left( {t,x_1 ,x_2 , \ldots ,x_n } \right) \mapsto u\left( {t,x_1 ,x_2 , \ldots ,x_n } \right) \hfill \\ \end{gathered} $$

Simulation and control of many dynamic systems involve solving partial differential equations (PDE). This letter proposes a semi-analytical solution (SAS) approach for fast and high-quality solution of first-order PDEs. The region of interest of the studied PDE is divided into a grid, and an SAS is derived for each grid cell in the form of the multivariate polynomials, of which the coefficients are identified using initial value and boundary value conditions. The solutions are solved in a “time-stepping” manner, i.e., within one time step, the coefficients of the SAS are identified and the initial value of the next time step is evaluated. This approach achieves a significantly larger grid cell than the widely used finite difference method, and thus enhances the computational efficiency significantly. The simulation result on the natural gas pipeline model demonstrates the advantages of SAS in accuracy and computational efficiency.

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.

Data movement between memory and processing units poses an energy barrier to Von-Neumann-based architectures. In-memory computing (IMC) eliminates this barrier. RRAM-based IMC has been explored for data-intensive applications, such as artificial neural networks and matrix-vector multiplications that are considered as “soft” tasks where performance is a more important factor than accuracy. In “hard” tasks such as partial differential equations (PDEs), accuracy is a determining factor. In this brief, we propose ReLOPE, a fully RRAM crossbar-based IMC to solve PDEs using the Runge–Kutta numerical method with 97% accuracy. ReLOPE expands the operating range of solution by exploiting shifters to shift input data and output data. ReLOPE range of operation and accuracy can be expanded by using fine-grained step sizes by programming other RRAMs on the BL. Compared to software-based PDE solvers, ReLOPE gains $31.4\times $ energy reduction at only 3% accuracy loss.

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lame—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.

For a first-order partial differential equation with the Dzhrbashyan – Nersesyan operator of fractional differentiation, we construct a fundamental solution and derive a general representation of the solutions in rectangular domains.

Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system.

Partial Differential Equations (PDE) appear in multiple Physic and Engineering applications. Normally, when modeling an application, the use of well-known and already solved PDE is considered. But what happens if a new PDE is used? Solving a new PDE is not an easy task. In this paper, we use a Computer Algebra System ( Cas ) in order to find the solution of PDE of first order. Specifically, we deal with Pfaff Equations, Quasilinear PDE and general first order PDE (using Lagrange–Charpit Method). To solve these PDE, we combine the power of a Cas with the flexibility of programming with it. Furthermore, the developed programs do not only provide the final result but also display all the intermediate steps which lead to find the solution of the PDE. This way, we introduce SFOPDES, a new Stepwise First Order PDE Solver which serves as a tutorial showing, step by step, the way to deal with PDE.

Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.

In this article, we investigate Neumann boundary value problem for Bitsadze equation in a ring domain by using same problem for the first order partial differential equations.

Grothendieck point residue is considered in the context of symbolic computation. Based on the theory of holonomic D-modules associated to a local cohomology class, a new effective method is given for computing Grothendieck point residue mappings. A basic strategy of our approach is the use of holonomic systems of first order linear partial differential equations. The resulting algorithm is easy to implement and can also be used to compute Grothendieck point residues in an effective manner.

Publisher Summary This chapter analyzes trigonometric Fourier series, which are series of trigonometric functions. This set is chosen because the mathematics is the best developed and is the simplest to demonstrate. It will serve to illustrate the basic questions that need to be addressed for each system. One technique that is often used in solving partial differential equations is separation of variables. This method yields a set of ordinary differential equations of which the solutions are pasted together to provide a solution to the partial differential equation. In the problems, each ordinary differential equation can be considered as an eigenvalue/eigenfunction problem where the differential operator is self-adjoint.

In 1984, H. Risken authored a book (H. Risken, The Fokker-Planck Equation: Methods of Solution, Applications, Springer-Verlag, Berlin, New York) discussing the Fokker-Planck equation for one variable, several variables, methods of solution and its applications, especially dealing with laser statistics. There has been a considerable progress on the topic as well as the topic has received greater clarity. For these reasons, it seems worthwhile again to summarize previous as well as recent developments, spread in literature, on the topic. The Fokker-Planck equation describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation. The structure of the Fokker-Planck equation for the vector case is

Galbrun's equation, which is a second order partial differential equation describing the evolution of a so-called Lagrangian displacement vector field, can be used to study acoustics in background flows as well as perturbations of astrophysical flows. Our starting point for deriving Galbrun's equation is linearized Euler's equations, which is a first order system of partial differential equations that describe the evolution of the so-called Eulerian flow perturbations. Given a solution to linearized Euler's equations, we introduce the Lagrangian displacement as the solution to a linear first order partial differential equation, driven by the Eulerian perturbation of the fluid velocity. Our Lagrangian displacement solves Galbrun's equation, provided it is regular enough and that the so-called "no resonance" assumption holds. In the case that the background flow is steady and tangential to the domain boundary, we prove existence, uniqueness, and continuous dependence on data of solutions to an initial--boundary value problem for linearized Euler's equations. For such background flows, we demonstrate that the Lagrangian displacement is well-defined, that the initial datum of the Lagrangian displacement can be chosen in order to fulfill the "no resonance" assumption, and derive a classical energy estimate for (sufficiently regular solutions to) Galbrun's equation. Due to the presence of zeroth order terms of indefinite signs in the equations, the energy estimate allows solutions that grow exponentially with time., Compared to the previous version some typos have been corrected and minor cosmetic changes have been made

A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system. Some applications to relevant partial differential equations are given., Comment: 16 pages, no figures

It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system polynomially homogeneous in the derivatives. The result is intimately related to the symmetry properties of the source system, and the proof, involving the use of the canonical variables associated to the admitted Lie point symmetries, is constructive. First order Monge-Amp\`ere systems, either with constant coefficients or with coefficients depending on the field variables, where the theorem can be successfully applied, are considered., Comment: 23 pages, no figures

This paper deals with Gevrey global solvability on the N-dimensional torus ( $${\mathbb {T}}^{N}\simeq {\mathbb {R}}^{N}/2\pi {\mathbb {Z}}^{N}$$ ) to a class of nonlinear first order partial differential equations in the form $$Lu-au-b{\overline{u}}=f$$ , where a, b, and f are Gevrey functions on $${\mathbb {T}}^{N}$$ and L is a complex vector field defined on $${\mathbb {T}}^{N}$$ . Diophantine properties of the coefficients of L appear in a natural way in our results. Also, we present results in $$C^\infty $$ context.

We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We study in detail a constant size growth model and a model with target size division. We also present versions of the model when the population is heterogeneous., Comment: 30 pages, 3 figures

We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Then, we exhibit regular quasisymmetric vector fields which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. The problems of boundary shape and locality are also discussed. We find that symmetric magnetic fields can be fitted into asymmetric domains, and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations., Comment: 19 pages, 5 figures

For families of hypersurfaces with singular points, a classical definition of an envelope is vague. In order to define an envelope for a family of hypersurfaces with singular points, we consider r-parameter families of frontals and of Legendre mappings in the unit tangent bundle over the Euclidean space. We define an envelope for the r-parameter family of Legendre mappings. Then the envelope is also a frontal. We investigate properties of the envelopes. As an application, we give a condition that the projection of a singular solution of a first order partial differential equation is an envelope.

Producción Científica, In this work, we study numerically a model which describes cell dwarfism. It consists in a pure initial value problem for a first order partial differential equation, that can be applied to the description of the evolution of diseases as thalassemia. We design two numerical methods that prevent the use of the characteristic curve x = 0, and derive their optimal rates of convergence. Numerical experiments are also reported in order to demonstrate the predicted accuracy of the schemes. Finally, a comparison study on their efficiency is presented., Junta de Castilla y León and European FEDER Funds (VA041P17), Junta de Castilla y León (VA138G18), Ministerio de Economía, Industria y Competitividad and European FEDER Funds (Project MTM2014-56022-C2-2-P), Ministerio de Ciencia e Innovación (Proyect MTM2017-85476-C2-P)

This study is concerned with a system of two nonlinear first order partial differential equations. The right-hand sides of the system contain the squares of the gradients of the unknown functions. Such type of Hamilton-Jacobi like equations are considered in mechanics and control theory. In the paper, we propose to search a solution in the form of an ansatz, the latter containing a quadratic dependence on the spatial variables and arbitrary functions of time. The use of this ansatz allows us to decompose the search of the solution’s components depending on the spatial variables and time. In order to find the dependence on the spatial variables one needs to solve an algebraic system of some matrix and vector equations and of a scalar equation. A general solution of this system of equations is found in a parametric form. To find the time-dependent components of the solution of the original system, we are faced with a system of nonlinear differential equations. The existence of exact solutions of a certain kind for the original system is established. A number of examples of the constructed exact solutions, including periodic in time and anisotropic in the spatial variables ones, are given. The spatial structure of the solutions is analyzed revealing that it depends on the rank of the matrix of the quadratic form entering the solution.

There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton’s algorithm linearizes the problem. An L 2 -residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.

The spherical Gardner (sG) equation is derived by reducing the (3+1) dimensional Gardner-Kadomtsev–Petviashvili (Gardner-KP) equation with a similarity reduction. As a special case, the step-like initial condition is considered along a paraboloid front. By applying a multiple-scale expansion, a system of first order partial differential equations for the slowly varying parameters of a periodic wavetrain is obtained. The corresponding modulation system is transformed into simpler form with the help of Riemann type variables. This basic form is important to investigate the dispersive shock wave (DSW) phenomena in the sG equation. DSW solution is also compared with the numerical solution of the sG equation and good agreement is found between these solutions.

The traditional wave equation is mostly, if not always, obtained from a system of first order partial differential equations augmented by constitutive relations. These are often nonlinear and linearizations are forcibly applied. In a nonlinear system of first order partial differential equations the criterion for hyperbolicity, necessary for the description of wave phenomena, involves the solution. It is therefore possible that solutions may evolve in such a way that hyperbolicity is challenged in the sense that the system comes close to not being hyperbolic. We use the recently introduced formulation for nonlinear acoustic disturbances to illustrate. When hyperbolicity deteriorates, standard numerical methods and the heuristics surrounding wave motion may be compromised. To overcome such difficulties we introduce the notion of inverse characteristicwhich, at least in the examples, reduces numerical calculations to elementary techniques and clarifies intuition. Analysis of inverse characteristics leads to two systems of ordinary differential equations that have time-like trajectories and space-varying associated curves. Time-like trajectories give rise to an alternative measure of time in terms of which space-like trajectories are easier to analyze. Space-varying curves enable the analysis of shock phenomena in a direct way. We give conditions under which an initially mild challenge of hyperbolicity, represented by pressure, develops into a severe challenge. Under these conditions violent velocity shocks develop from an initially undisturbed state.

A quasi-linear differential algebraic system of partial differential equations with a special structure of the pencil of Jacobian matrices of small index is considered. A nonlinear spline collocation difference scheme of high approximation order is constructed for the system by approximating the required solution by a spline of arbitrary in each independent variable. It is proved by the simple iteration method that the nonlinear difference scheme has a solution that is uniformly bounded in the grid space. Numerical results are demonstrated using a test example.

In this paper, a numerical method is proposed to estimate the solution of initial-boundary value problems for a class of partial integro-differential equations. This is based on the cubic B-splines method for spatial derivatives while the backward Euler method is used to discretize the temporal derivatives. Detailed discrete schemes are investigated. Next, we proved the convergence and the stability of the proposed method. The method is applied to some test examples and the numerical results have been compared with the exact solutions. The obtained results show the computational efficiency of the method. It can be concluded that computational efficiency of the method is effective for the initial-boundary value problems. Keywords: Cubic B-splines, Partial integro-differential equation, Backward Euler method

We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions., Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/

International audience; In this paper we consider the axial motion of a drill-string and the interaction of the drill-bit with the formation. We design an observer that estimates, in real time, the axial speed and force along the drill-string and at the drill-bit using only topside measurements (force and velocity). More generally, our approach enables the design of robust observers for systems of Ordinary Differential Equation coupled with a first order Partial Differential Equation in their actuation path (a class of systems to which belongs the considered drill-string axial dynamics). However, such an observer requires the knowledge of different physical parameters among which the rock intrinsic energy which is a priori unknown. Thus, we combine our observer with an algorithm that provides a near real-rime estimation of the seismic velocities of rocks interacting with the drill-bit, using seismic-while-drilling. The efficiency of the proposed approach is shown through simulation results.

Exponentially stable periodic orbits of ordinary differential equations and their basins' of attraction are characterized by contraction metrics. The advantages of a contraction metric over a Lyapunov function include its insensitivity to small perturbations of the dynamics and the exact location of the periodic orbit. We present a novel algorithm to rigorously compute contraction metrics, that combines the numerical solving of a first order partial differential equation with rigorous verification of the conditions for a contraction metric. Further, we prove that our algorithm is able to compute a contraction metric for any ordinary differential equation possessing an exponentially stable periodic orbit. We demonstrate the applicability of our approach by computing contraction metrics for three systems from the literature.

We show that the Mellin transform of an $n$-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of $(n-2)$ linear first order partial differential equations corresponding to $(n-2)$ positive helicity gluons. Although these equations closely resemble Knizhnik-Zamolodchikov equations for $SU(N)$ current algebra there is also an additional "correction" term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Similar equations can also be written down for the momentum space tree level MHV scattering amplitudes. We also propose a way to deal with the non closure of subleading current algebra generators under commutation. This is then used to compute some subleading terms in the mixed helicity gluon OPE and our results match with those obtained from an explicit calculation using the Mellin MHV amplitude., 26+23 Pages, Latex, Typo corrected, Some discussions on subleading soft gluon symmetry is changed, Results unchanged

This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in $${\mathbb {R}}^d$$ in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on $${\mathbb {R}}^d$$ equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem.

In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.

Second order stochastic partial differential equations are discussed from a rough path point of view. In the linear and finite-dimensional noise case we follow a Feynman–Kac approach which makes good use of concentration of measure results, as those obtained in Sect. 11.2. Alternatively, one can proceed by flow decomposition and this approach also works in a number of non-linear situations. Secondly, now motivated by some semi-linear SPDEs of Burgers’ type with infinite-dimension noise, we study the stochastic heat equation (in space dimension 1) as evolution in Gaussian rough path space relative to the spatial variable, in the sense of Chap. 10.

This study aims to review the physical theory and parametrizations associated to Turbulent Kinetic Energy Density Function (STKE). The bibliographic references bring a broad view of the physical problem, mathematical techniques and modeling of turbulent kinetic energy dynamics in the convective boundary layer. A simplified model based on the dynamical equation for the STKE, in an isotropic and homogeneous turbulent flow regime, is done by formulating and considering the isotropic inertial energy transfer and viscous dissipation terms. This model is described by the Cauchy Problem and solved employing the Method of Characteristics. Therefore, a discussion on Linear First Order Partial Differential Equation, its existence, and uniqueness of solution has been presented. The spectral function solution obtained from its associated characteristic curves and initial condition (Method of Characteristics) reproduces the main features of a modeled physical system. In addition, this modeling allows us to obtain the scaling parameters, which are frequently employed in parameterizations for turbulent dispersion.

In this study, the author used the joint Fourier- Laplace transform to solve non-homogeneous time fractional first order partial differential equation with non-constant coefficients. Constructive examples are also provided throughout the paper. It is a remarkable feature of the first order fractional differential equations that a procedure can be developed for solving this equation, regardless of its complexity.

An approach of fabricating Parity-Time symmetry electronic dimers circuits is demonstrated using an imaginary resistor Z whose impedance is frequency independent. Remarkably, the dynamic rate equations of the coupled oscillators so-called ZRC dimers, uniquely consist of first order partial differential equations which result in a 2 ×2 effective Hamiltonians commuting with the joint Parity-Time (PT) operator. The method helps in describing two groups of PT symmetry electronics. The first group has an eigenspectrum that displays a single spontaneous PT transition or a thresholdless (THL) PT transition. The second group is a PT transitionless normal mode spectrum with essentially negative values of capacitors. The dynamics indicate that the system exhibits high frequencies propagation when operation is done before the thresholdless point is reached. The transitionless regime indicates an equilibrium point in the coupling for which the system turns to be conservative regardless the gain/loss values, and beyond which the gain/loss remarkably decays/amplifies wave. In addition, we observe a value of gain/loss above which there is a limitation of signal amplification.

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(\alpha,\beta)$-Finsler spaces, respectively $(A,B)$-Finsler spacetimes, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the defining $1$-form. We illustrate our results with novel examples of $(\alpha,\beta)$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries., Comment: 17 pages, results on $(\alpha,\beta)$-Finsler geometries extended, explicit examples added, updated to journal version