Nonlinear Second-order Partial Differential Equations Research Articles

Worst-Case Portfolio Optimization with Proportional Transaction Costs

We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be in either a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario.We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.

A differential equation for the Generalized Born radii

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.

Dynamic Mean-Variance Model with Borrowing Constraint under the Constant Elasticity of Variance Process

This paper studies a continuous-time dynamic mean-variance portfolio selection problem with the constraint of a higher borrowing rate, in which stock price is governed by a constant elasticity of variance (CEV) process. Firstly, we apply Lagrange duality theorem to change an original mean-variance problem into an equivalent optimization one. Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Furthermore, we use Legendre transform and dual theory to transform the HJB equation into its dual one. Finally, the closed-form solutions to the optimal investment strategy and efficient frontier are derived by applying variable change technique.

Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

Solvability of concatenated Runge–Kutta equations for second-order nonlinear PDEs

In this paper, we consider the solvability of concatenated Runge–Kutta (CRK) equations, which are the nonlinear algebraic equations arising from CRK methods applied to some kind of second-order nonlinear partial differential equations. To guarantee the existence and uniqueness of the solution of CRK equations, temporal and spatial step-sizes of CRK methods should be chosen appropriately. Under different assumptions, with the homotopy continuation method, we choose different step-sizes to ensure the solvability of CRK equations.

Wavefront solutions of a nonlinear telegraph equation

The paper is concerned with a class of nonlinear second-order hyperbolic partial differential equations. Necessary and sufficient conditions for the occurrence of traveling-wave solutions are derived, and the spectra of wave-speeds for which such solutions exist are characterized. An integral equation approach is employed. Models of branching random walks, conduction in transmission lines, and biological dispersal provide illustrations.

On the density of systems of non-linear spatially homogeneous SPDEs

In this paper, we consider a system of k second-order nonlinear stochastic partial differential equations with spatial dimension , driven by a q-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and Hölder's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of stochastic partial differential equations (SPDEs). In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span . We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span which contains this point has a non-void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension .

Computational optogenetics: A novel continuum framework for the photoelectrochemistry of living systems

Electrical stimulation is currently the gold standard treatment for heart rhythm disorders. However, electrical pacing is associated with technical limitations and unavoidable potential complications. Recent developments now enable the stimulation of mammalian cells with light using a novel technology known as optogenetics. The optical stimulation of genetically engineered cells has significantly changed our understanding of electrically excitable tissues, paving the way towards controlling heart rhythm disorders by means of photostimulation. Controlling these disorders, in turn, restores coordinated force generation to avoid sudden cardiac death. Here, we report a novel continuum framework for the photoelectrochemistry of living systems that allows us to decipher the mechanisms by which this technology regulates the electrical and mechanical function of the heart. Using a modular multiscale approach, we introduce a non-selective cation channel, channelrhodopsin-2, into a conventional cardiac muscle cell model via an additional photocurrent governed by a light-sensitive gating variable. Upon optical stimulation, this channel opens and allows sodium ions to enter the cell, inducing electrical activation. In side-by-side comparisons with conventional heart muscle cells, we show that photostimulation directly increases the sodium concentration, which indirectly decreases the potassium concentration in the cell, while all other characteristics of the cell remain virtually unchanged. We integrate our model cells into a continuum model for excitable tissue using a nonlinear parabolic second-order partial differential equation, which we discretize in time using finite differences and in space using finite elements. To illustrate the potential of this computational model, we virtually inject our photosensitive cells into different locations of a human heart, and explore its activation sequences upon photostimulation. Our computational optogenetics tool box allows us to virtually probe landscapes of process parameters, and to identify optimal photostimulation sequences with the goal to pace human hearts with light and, ultimately, to restore mechanical function.

Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions

We are concerned with the solvability of nonlinear second-order elliptic partial differential equations with nonlinear boundary conditions. We study the generalized Steklov–Robin eigenproblem (with possibly singular weights) in which the spectral parameter is both in the differential equation and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates.

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.

Estimates of solutions for parabolic differential and difference functional equations and applications

The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the com- parison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. The aim of the paper is to prove theorems on the estimates of solutions for non- linear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for generated by them implicit finite difference functional schemes. We also give the applications of the results. More precisely, we prove the theorem on the convergence of a difference method to a classical solution for the differential functional problem, which by the given estimates, may be treated in the subspace C (,R) ⊂ C (,R), where R ⊂ R is an interval. It is a new idea in area of nonlinear implicit difference methods which was studied for explicit methods by K. Kropielnicka and L. Sapa (14). This considerably extends the class of problems which are solvable by the described method. Therefore, the Lipschitz, Perron or generalized Perron conditions posed onf with respect toz need not be global, inC (,R), as in the papers due to M. Malec, Cz. Mączka, W. Voigt, M. Rosati and L. Sapa (15-19,24,25),

Approximate analytical solutions for a mathematical model of a tubular packed-bed catalytic reactor using an Adomian decomposition method

In this paper, a mathematical model of a tubular packed-bed catalytic reactor, which is modeled by a system of strongly nonlinear second-order partial differential equations with incompatible boundary conditions, will be solved. By properly using the boundary conditions and correctly choosing the solution search direction, approximate analytic solutions for the model can be obtained by the Adomian decomposition method. When the values of the dimensionless parameters in the system are assigned within a suitable range, the solutions describe objectively the distributions of the temperature and key reactant concentration in the reactor.

Modeling of Placement of Immiscible Fluids of Different Rheology into a Hydraulic Fracture

A model of consecutive placements of two different immiscible fluids, possessing different densities and rheological properties, into an expanding fracture is presented. Hydraulic fracturing technology, based on such a fluid placement, is expected to allow better fracture shape control. Mathematically, the problem is reduced to solving a two-dimensional second-order nonlinear partial differential equation describing pressure distribution in a semiellipsoid domain with moving boundaries. The level set method is used to track the interface between the fluids in time. Numerical examples of the placement of power-law fluids are presented, and the accuracy of the computations is investigated.

A Monge-Ampère enhancement for semi-Lagrangian methods

Demanding the compatibility of semi-Lagrangian trajectory schemes with the fundamental Euler expansion formula leads to the Monge-Ampère (MA) nonlinear second-order partial differential equation. Given standard estimates of the departure points of flow trajectories, solving the associated MA problem provides a corrected solution satisfying a discrete Lagrangian form of the mass continuity equation to round-off error. The impact of the MA enhancement is discussed in two diverse limits of fluid dynamics applications: passive tracer advection in a steady cellular flow and in fully developed turbulence. Improvements of the overall accuracy of simulations depend on the problem and can be substantial.

SOPDEs and nonlinear connections

The canonical k-tangent structure on $T^1_kQ=TQ\oplus\stackrel{k}...\oplus TQ$ allows us to characterize nonlinear connections on $T^1_kQ$ and to develop G\"unther's (k-symplectic) Lagrangian formalism. We study the relationship between nonlinear connections and second-order partial differential equations (SOPDEs), which appear in G\"unther's Lagrangian formalism.

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.

A Dynamic Semi-Analytic Channel-to-Channel Model of Two-Phase Water Distribution for a Unit Fuel Cell

The critical task of controlling the water accumulation within the gas diffusion layer (GDL) and the channels of a polymer-electrolyte-membrane (PEM) fuel cell is shown to benefit from a partial-differential-equation (PDE) approach. Starting from first principles, a model of a fuel cell is represented as a boundary value problem for a set of three coupled nonlinear second-order PDEs for mass transport across the GDL of each electrode. These three PDEs are approximated, with justification founded in linear systems theory and a time-scale decomposition approach, by a semianalytic model that requires less than one-third the number of states to be numerically integrated. A set of numerical transient, analytic transient, and analytic steady-state solutions for the semianalytic model are presented, and an experimental verification of the cell voltage prediction due to liquid-water accumulation is demonstrated. The semianalytic model derived and the associated analysis represent our main contribution for which future expansion of along-the-channel dynamics and statistical consideration of cell-to-cell variations can be implemented for application to control, estimation, and diagnostic algorithms.

Development of a Numerical Method for Analyzing Fire Plume Equations Using an Improved Version of a Quasi-Third-Order Accurate CIP Method That Eliminates Numerical Oscillation

This study is concerned with simulations of fire phenomena using field equation models. When performing numerical computation of a fire plume accompanied by gas and smoke, the phenomena are expressed through simultaneous non-stationary nonlinear second-order partial differential equations containing advective terms. Those advective terms are then subjected to difference approximations. The ordinary CIP method, which allows us to find an approximate solution using a third-order interpolation function, is often influenced by numerical oscillations emerging specifically with the third-order accuracy, thereby drastically reducing the numerical stability. We developed a new computational algorithm called mCIP method to eliminate these numerical oscillations generated in the numerical computation of partial differential equations when a third-order accurate method is used. A characteristic feature of this mCIP is that it is based on the first-order upwind difference scheme, which does not generate numerical oscillations in those regions where such oscillations may normally occur. We successfully evaluated the performance of this mCIP method and verified its effectiveness by comparing the results of computations with other representative methods, such as CIP method, TVD method and first-order accurate difference method.

On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data

In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.