Linear Differential Equations Research Articles

An analytical solution for vertical infiltration in homogeneous bounded profiles

AbstractIn this study, we derive an analytical solution to address the problem of vertical infiltration within 1D homogeneous bounded profiles. Initially, we consider the Richards equation together with Dirichlet boundary conditions. We assume constant diffusivity and linear dependence between the conductivity and the water content, resulting to a linear partial differential equation of diffusion type. To solve the simplified initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation of the solution that can be efficiently and directly computed numerically, yielding a convergent scheme. We examine various cases, and we compare our solution with well‐known approximate solutions. This work can be seen as a first step to derive analytical solutions for the far more difficult and complex problem of modelling water flow in heterogeneous layered soils.

Decoupling a System of Linear Differential Equations into Blocks

Decoupling a System of Linear Differential Equations into Blocks

Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method.

Solving Advanced Missile Robotic Control Problem Via Integral Rohit transform

The problem of robotic control of an advanced missile requires designing control systems for remote-controlled missiles with sufficient abilities. These control systems need to handle navigation, guidance, propulsion, and potentially other tasks like target acquisition and avoidance of obstacles. The various aspects of missile dynamics are described by complex and non-linear differential equations. Due to this, it becomes difficult to obtain their analytical solutions. During system design and analysis, to solve these complex and non-linear differential equations, numerical techniques like Euler’s technique, the Runge-Kutta technique, or more advanced mathematical techniques are generally used. The advanced missile robotic control problem is one of the problems represented by second-order linear differential equations. In this paper, the integral Rohit transform (RT) has been applied to solve an advanced missile robotic control problem. This transformation has never been applied in working out the advanced missile robotic control problem. This paper applies a new technique for solving the second-order linear differential equation representing the advanced missile robotic control problem to obtain its required turn angle so that it will be able to catch and destroy the attacker. The necessary graphs are plotted to illustrate the numerical solution of the robotic control of an advanced missile. It reveals that the integral Rohit transform (RT) technique is an effective technique to solve the advanced missile robotic control problem.

Generalized periodicity and applications to logistic growth

Classically, a continuous function f:R→R is periodic if there exists an ω>0 such that f(t+ω)=f(t) for all t∈R. The extension of this precise definition to functions f:Z→R is straightforward. However, in the so-called quantum case, where f:qN0→R (q>1), or more general isolated time scales, a different definition of periodicity is needed. A recently introduced definition of periodicity for such general isolated time scales, including the quantum calculus, not only addressed this gap but also inspired this work. We now return to the continuous case and present the concept of ν-periodicity that connects these different formulations of periodicity for general discrete time domains with the continuous domain. Our definition of ν-periodicity preserves crucial translation invariant properties of integrals over ν-periodic functions and, for ν(t)=t+ω, ν-periodicity is equivalent to the classical periodicity condition with period ω. We use the classification of ν-periodic functions to discuss the existence and uniqueness of ν-periodic solutions to linear homogeneous and nonhomogeneous differential equations. If ν(t)=t+ω, our results coincide with the results known for periodic differential equations. By using our concept of ν-periodicity, we gain new insights into the classes of solutions to linear nonautonomous differential equations. We also investigate the existence, uniqueness, and global stability of ν-periodic solutions to the nonlinear logistic model and apply it to generalize the Cushing–Henson conjectures, originally formulated for the discrete Beverton–Holt model.

On the dimension of the solution space of linear difference equations over the ring of infinite sequences

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information on the dimension is available.

Flow analysis of temperature-dependent variable viscosity Phan Thien Tanner fluid thin film over a horizontally moving heated plate

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

Dense outputs from quantum simulations

The quantum dense output problem is the process of evaluating time-accumulated observables from time-dependent quantum dynamics using quantum computers. This problem arises frequently in applications such as quantum control and spectroscopic computation. We present a range of algorithms designed to operate on both early and fully fault-tolerant quantum platforms. These methodologies draw upon techniques like amplitude estimation, Hamiltonian simulation, quantum linear Ordinary Differential Equation (ODE) solvers, and quantum Carleman linearization. We provide a comprehensive complexity analysis with respect to the evolution time T and error tolerance ϵ. Our results demonstrate that the linearization approach can nearly achieve optimal complexity O(T/ϵ) for a certain type of low-rank dense outputs. Moreover, we provide a linearization of the dense output problem that yields an exact and finite-dimensional closure which encompasses the original states. This formulation is related to the Koopman Invariant Subspace theory and may be of independent interest in nonlinear control and scientific machine learning.

A note on the inverse problem for finite differential Galois groups

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to H72, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of PSL3(ℂ) are the projective monodromy of a hypergeometric equation.

Some properties of solutions of a linear set-valued differential equation with conformable fractional derivative

The article explores a linear set-valued differential equation featuring both conformable fractional and generalized conformable fractional derivatives. It presents conditions for the existence of solutions and provides analytical expressions for the shape of solution sections at different time points. Model examples are employed to illustrate the results.

Data-driven identification of stable sparse differential operators using constrained regression

Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics forecasting problems. In this article, we propose a novel methodology for learning differential operators that are theoretically linearly stable and have sparsity patterns of common discretization schemes. These differential operators are obtained by solving a constrained regression problem, involving local constraints to ensure the linear stability of the global dynamical system. We further extend this approach for learning nonlinear differential operators by determining linear stability constraints for linearized equations around an equilibrium point. The applicability of the proposed method is demonstrated for both linear and nonlinear partial differential equations such as 1-D scalar advection-diffusion equation, 1-D Burgers equation and 2-D advection equation. The results indicated that solutions to constrained regression problems with linear stability constraints provide accurate and linearly stable sparse differential operators.

Subharmonic functions with separated variables and their connection with generalized convex functions

In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables, representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, i.e., convex with respect to two functions.

Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.

Stability analysis of linear fractional neutral delay differential equations

Stability analysis of linear fractional neutral delay differential equations

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Deep learning based on randomized quasi-Monte Carlo method for solving linear Kolmogorov partial differential equation

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations (PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization (ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is O(n−1+ϵ) for arbitrarily small ϵ>0, while for the MC method it is O(n−1/2+ϵ) for arbitrarily small ϵ>0. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as n increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative L2 error than that based on the MC method.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

D-module techniques for solving differential equations in the context of Feynman integrals

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare D-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic D-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.

Spectral moments of the real Ginibre ensemble

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments M2pr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{2p}^\ extrm{r}$$\\end{document}. The latter are expressed in terms of the 3F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_3 F_2$$\\end{document} hypergeometric functions, with a simplification to the 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2 F_1$$\\end{document} hypergeometric function possible for p=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=0$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence {M2pr}p=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{ M_{2p}^\ extrm{r} \\}_{p=0}^\\infty $$\\end{document}. Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.

The BVP of a class of second order linear fuzzy differential equations is solved by Green function method under the concept of granular differentiability

The BVP of a class of second order linear fuzzy differential equations is solved by Green function method under the concept of granular differentiability