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In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving L 2 -contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients. [ABSTRACT FROM AUTHOR]

In this study, we addressed the nonoscillation of the Sturm--Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm--Liouville type. [ABSTRACT FROM AUTHOR]

In this paper, we studied the linear 2-refined neutrosophic differential equations and defined the homogeneous and non-homogeneous 2-refined neutrosophic differential equations. Also to presenting 2-refined neutrosophic differential equation of Bernoulli, which turns into a linear equation and thus facilitates its solution. Apart from talking about how to solve these equations and giving enough examples to support it. [ABSTRACT FROM AUTHOR]

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard's iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is the treatmnent of the theory of linear ordinary differential equations by the well-known traditional methods of series and iterations. In the iteration method, instead of the continuity we require the analyticity of the coefficients of the differential equations. Then the iterations can be carried out for the arbitrary form of the coefficients. Our procedure determines the fundamental system of particular integrals {y1, Y2, ..., Yn} with arbitrary precision. Therefore our theory of differential equations is non-Liouville. (In the Liouville theory we know only that the solution has the form y(x) = L-=o CkYk, without a procedure for finding the particular integrals). The Wronskian W(y1, Y2, ..., Yn) = 1!2! • • • (n - 1)! of the fundamental system of a differential equation of n-th order has a new meaning. It becomes the basic metric relation in the space of particular solutions. Knowing that linear differntial equations of the second order with constant coefficients correspond to usual trigonometry there exists a natural generalization of the trigonometry (elliptical and hyperbolical). From the Wronskian as a basic trigonometrical relation the Euler formula, addition formulae etc. follow. The solutions in the form of a series of multiple integrals provide a natural approach to approximate calculations. The approximate solutions are especially suitable in the case of monotone coefficients of differential equations. Working in the space of the fundamental particular integrals Y1 ® Y2 ® • • • ® Yn one can obtain interesting geometrical relations. [ABSTRACT FROM AUTHOR]

Three new approaches to the solution of the Bratu problem are presented. The first approach realizes the idea of successive differentiating the initial equation of this problem with expansion of the sought for function at the symmetry point of a space. The second approach is associated with the additional integration of the differential Bratu equation, and it represents a hybrid integral method. The third approach is based on the combined application of the successive differentiation of the Bratu equation, the hybrid integral method, the expansion of the sought for function at two points of the space, and an additional integral relation. The indicated approaches are characterized by the simplicity of the calculations required for their realization, and they reduce the solution of the Bratu problem to the solution of a system of linear algebraic equations. The numerical results of solving this problem demonstrate the high efficiency of the approaches proposed, providing the obtaining of the solutions whose accuracy exceeds the accuracy of the analogous solutions, obtained on the basis of the known numerical and numerical-analytical methods, by three to five orders of magnitude. This is especially true for the third approach that allows one to obtain two classical solutions of the Bratu problem fairly simply and with a very high accuracy. It is shown that the obtaining of an approximate solution of the Bratu problem with this approach calls for a small number of series terms, and the solutions obtained converge quickly and approximate the problem highly exactly. [ABSTRACT FROM AUTHOR]

Systems of fractional order differential and pseudo-differential equations are used in modeling of various dynamical processes. In the analysis of such models, including stability analysis, asymptotic behaviors, etc., it is useful to have a representation formulas for the solution. In this paper we prove the existence and uniqueness theorems and derive representation formulas for the solution of general systems of fractional multi-order linear pseudo-differential equations through the matrix-valued Mittag-Leffler function. Examples illustrating the obtained results are also provided. [ABSTRACT FROM AUTHOR]

For a given direction b ∈ C n \ {0} we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of L-index in the direction with a positive continuous function L satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions as the coefficients of the equation. Our considerations use some estimates involving a directional logarithmic derivative and distribution of zeros on all directional slices in the unit ball. [ABSTRACT FROM AUTHOR]

For one generalization of the Riemann differential equation, we obtain sufficient conditions for the approximability by twice continuously differentiable rational interpolation spline functions. To solve the corresponding boundary value problem numerically, a tridiagonal system of linear algebraic equations is constructed and conditions on the coefficients of the differential equation are found guaranteeing the uniqueness of the solution of such Estimates of the deviation of the discrete solution of the boundary value problem from the exact solution on a grid are presented. [ABSTRACT FROM AUTHOR]

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. [ABSTRACT FROM AUTHOR]

In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context. [ABSTRACT FROM AUTHOR]

The authors show that any semicanonical or noncanonical third-order linear delay dynamic equation on a time scale can be written in canonical form without imposing any additional conditions on the coefficient functions. Since this is true for any time scale, this means it holds for differential and difference equations. The implication of this is the significant result that any set of conditions which show that a related equation in canonical form is oscillatory will guarantee that the semicanonical or noncanonical equation is also oscillatory. Several examples of the application of the results are incorporated into the paper. [ABSTRACT FROM AUTHOR]

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin's algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems. [ABSTRACT FROM AUTHOR]

Sufficient conditions are established for the boundedness of all solutions and their first two derivatives of a third-order linear integro-differential equation of the Volterra type on the half-line. To this end, using a method proposed by the first author in 2006, first, we reduce the equation under consideration to an equivalent system consisting of one first-order differential equation and one second-order Volterra integro-differential equation. Then a new generalized Lyapunov functional is proposed for this system, the nonnegativity of this functional on solutions of this system is proved, and an upper bound is given for the derivative of this functional via the original functional. The resulting estimate is an integro-differential inequality whose solution gives an estimate of the functional. [ABSTRACT FROM AUTHOR]

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation's solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties. [ABSTRACT FROM AUTHOR]

We establish sharp relations between several regularity coefficients for flows generated by non-autonomous differential equations. We verify that these relations are sharp by showing that all possible values of the regularity coefficients satisfying these relations are attained by some linear equation with bounded piecewise-continuous coefficient matrix. In addition, we introduce two new regularity coefficients and we obtain sharp relations between these and other coefficients. [ABSTRACT FROM AUTHOR]

Isothermal bainitic transformation is studied in commercial and experimental alloy steels of various compositions using dilatometry. The effect of the conditions of cooling to the temperature of isothermal quenching on the subsequent kinetics of the bainitic transformation is considered. The experimental kinetics of formation of bainite is described with the help of the Austin–Rickett (AR) equation. The temperature dependence of parameters n and ln (k) in the AR equation and their linear interrelationship are presented. It is shown that the dependences n = f [ln (k)] differ for steels of specific classes, and the difference may be associated with different stabilities of the supercooled austenite. It is shown that in a number of cases the kinetics of the isothermal bainitic transformation is describable adequately by the Kolmogorov–Johnson–Mehl–Avrami (KJMA) equation. The suggested differential equation generalizing the AR and KJMA equations makes it possible to describe the kinetics of formation of bainite the most accurately. [ABSTRACT FROM AUTHOR]

The main objective of this article is to discuss a numerical method for solving singular differential equations based on wavelets. Singular differential equations are first transformed into a system of linear algebraic equations, and then the linear system’s solution produces the unknown coefficients. Along with its estimated error, the convergence of the approximative solution is also determined. Some numerical examples are thought to show that Bernoulli wavelet is better than Chebyshev and Legendre wavelet and other existing techniques. [ABSTRACT FROM AUTHOR]

We consider a scalar linear mixed differential equation with several terms, both delayed and advanced arguments and a bounded right-hand side. Assuming that the deviations of the argument are bounded, we present sufficient conditions when there exists a unique bounded solution on the positive half-line. Explicit tests for a bounded solution of a homogeneous equation to decay exponentially are obtained. Existence of exponentially decaying solutions for this class of differential equations is studied for the first time, and we illustrate sharpness of the results with examples. We show that the standard approach when convergence of all solutions is stated does not work for mixed equations; in addition to an exponentially decaying, there may be a growing solution. All the coefficients and the mixed arguments are assumed to be Lebesgue measurable functions, not necessarily continuous. Though the equation is linear, some properties, as well as the methods applied, are more typical for nonlinear models, for example, fixed-point theorems are used in the proofs. [ABSTRACT FROM AUTHOR]

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields [1] with the rough calculus on manifolds developed by Cass, Litterer and Lyons in [2]. [ABSTRACT FROM AUTHOR]

In this paper, we, for the first time, prove the local solvability and stability of an inverse spectral problem for higher-order ( n > 3 ) differential operators with distribution coefficients. The inverse problem consists of the recovery of differential equation coefficients from (n − 1) spectra and the corresponding weight numbers. The proof method is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation remains uniquely solvable. Furthermore, we estimate the differences of the coefficients in the corresponding functional spaces. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce interval two-point fuzzy boundary value problems, in which the fuzziness course when the coefficients of the governing ordinary differential equation and/or the boundary conditions include fuzzy numbers of either triangular or trapezoidal types. Such equations will be solved by introducing the concept of α – level sets, α  [0,1] to treat the fuzzy ordinary differential equation into two nonfuzzy ordinary differential equations, which correspond to the lower and upper solutions of the interval fuzzy solutions. The well-known variational iteration method has been used to solve two-point fuzzy boundary value problems and linear equations have been examined. [ABSTRACT FROM AUTHOR]

In this paper, we analyze oscillatory properties of perturbed half‐linear differential equations (i.e., equations with one‐dimensional p‐Laplacian). The presented research covers the Euler and Riemann–Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well. [ABSTRACT FROM AUTHOR]

A new method is proposed for reducing spark range data using a Gauss pseudospectral collocation. Existing methods use numerical integration to propagate the differential equation model and its sensitivities forward in time. Here, a collocation method is used to transcribe the differential equations into algebraic ones that are easily solved. Realizing that the auxiliary equations used to find model sensitivities are linear, a second transcription renders a set of linear algebraic equations that solve exactly for the needed sensitivities without iteration. Thus, measurement sensitivities with respect to initial conditions and aerodynamic parameters are easily found through solution of a set of linear algebraic equations. The method is demonstrated on a set of actual data from the M898 155-mm projectile. For models using fixed-plane coordinates, and only the 10 most influential aerodynamic coefficients, the new method produces results that rival established commercial codes in accuracy. [ABSTRACT FROM AUTHOR]

We study the relation between the Galois group G of a linear difference-differential system and two classes C 1 and C 2 of groups that are the Galois groups of the specializations of the linear difference equation and the linear differential equation in this system respectively. We show that almost all groups in C 1 ∪ C 2 are algebraic subgroups of G , and there is a nonempty subset of C 1 and a nonempty subset of C 2 such that G is the product of any pair of groups from these two subsets. These results have potential application to the computation of the Galois group of a linear difference-differential system. We also give a criterion for testing linear dependence of elements in a simple difference-differential ring, which generalizes Kolchin's criterion for partial differential fields. [ABSTRACT FROM AUTHOR]

Radial basis functions are typically used when discretization schemes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrix-matrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is locally conservative, keeping the error virtually constant throughout the computation. [ABSTRACT FROM AUTHOR]

In this study, the slope deflection method was presented for structures made of small-scaled axially functionally graded beams with a variable cross section within the scope of nonlocal elasticity theory. The small-scale effect between individual atoms cannot be neglected when the structures are small in size. Therefore, the theory of nonlocal elasticity is used throughout. The stiffness coefficients and fixed-end moments are calculated using the method of initial values. With this method, the solution of the differential equation system is reduced to the solution of the linear equation system. The given transfer matrix is unique and the problem can be easily solved for any end condition and loading. In this problem, double integrals occur in terms of the transfer matrix. However, this form is not suitable for numerical calculations. With the help of Cauchy's repeated integration formula, the transfer matrix is given in terms of single integrals. The analytical or numerical calculation of single integrals is easier than the numerical or analytical calculation of double integrals. It is demonstrated that the nonlocal effect plays an important role in the fixed-end moments of small-scaled beams. [ABSTRACT FROM AUTHOR]

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation y′′+λ/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm. [ABSTRACT FROM AUTHOR]

We consider a second-order nonlinear wave equation with a linear convolution term. When the convolution operator is taken as the identity operator, our equation reduces to the classical elasticity equation which can be written as a p-system of first-order differential equations. We first establish the local well-posedness of the Cauchy problem. We then investigate the behavior of solutions to the Cauchy problem in the limit as the kernel function of the convolution integral approaches to the Dirac delta function, that is, in the vanishing dispersion limit. We consider two different types of the vanishing dispersion limit behaviors for the convolution operator depending on the form of the kernel function. In both cases, we show that the solutions converge strongly to the corresponding solutions of the classical elasticity equation. [ABSTRACT FROM AUTHOR]

The q-homotopy analysis transform method (q-HATM) is a powerful tool for solving differential equations. In this study, we apply the q-HATM to compute the numerical solution of the fractional-order Kolmogorov and Rosenau–Hyman models. Fractional-order models are widely used in physics, engineering, and other fields. However, their numerical solutions are difficult to obtain due to the non-linearity and non-locality of the equations. The q-HATM overcomes these challenges by transforming the equations into a series of linear equations that can be solved numerically. The results show that the q-HATM is an effective and accurate method for solving fractional-order models, and it can be used to study a wide range of phenomena in various fields. [ABSTRACT FROM AUTHOR]

In this paper, the fuzzy approximate solutions for the fuzzy Hybrid differential equation emphasizing the type of generalized Hukuhara differentiability of the solutions are obtained by using the two-dimensional Müntz–Legendre wavelet method. To do this, the fuzzy Hybrid differential equation is transformed into a system of linear algebraic equations in a matrix form. Thus, by solving this system, the unknown coefficients are obtained. The convergence of the proposed method is established in detail. Numerical results reveal that the two-dimensional Müntz–Legendre wavelet is very effective and convenient for solving the fuzzy Hybrid differential equation. [ABSTRACT FROM AUTHOR]

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved. [ABSTRACT FROM AUTHOR]

In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and we obtain a family of linearized equations. By making use of the generalized shifted airfoil polynomials of the second kind (SAPSK) together with some appropriate collocation points as the roots of SAPSK, we arrive at an algebraic system of linear equations to be solved in an iterative manner. The error analysis and convergence properties of the SAPSK are established in the L 2 and L ∞ norms. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is not only capable of tackling the inherit singularity at the origin, but also produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model. The computational outcomes are accurate in comparison with those of numerical values reported in the literature. [ABSTRACT FROM AUTHOR]

We derive the existence, uniqueness, and uniform estimates for the abstract Navier–Stokes problem with small parameters in half-space. The equation involves small parameters and an abstract operator in a Banach space . Hence, we obtain the singular perturbation property for the Stokes operator depending on a parameter. We can obtain the various classes of Navier–Stokes equations by choosing and the linear operators . These classes occur in a wide variety of physical systems. As application we establish the existence, uniqueness, and uniform estimates for the solution of the mixed problems for infinitely many Navier–Stokes equations and nonlocal mixed problems for the high order Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Copyright of German International Journal of Modern Science / Deutsche Internationale Zeitschrift für Zeitgenössische Wissenschaft is the property of Artmedia24 and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The oscillation property of the second order half linear dynamic equation was studied, some sufficient conditions were obtained to ensure the oscillation of all solutions of the equation. The results are supported by illustrative examples. [ABSTRACT FROM AUTHOR]

Consider a Lotka–Volterra (LV) system of coupled differential equations: x ̇ k = x k (r k − x k + (B x) k) , x = (x k) , 1 ≤ k ≤ n , where r = (r k) is a n × 1 vector and B a n × n matrix. Assume that the interaction matrix B is random and follows the elliptic model: B = 1 α n A + μ n 1 n 1 n T , where A = (A i j) is a n × n matrix with N (0 , 1) entries satisfying the following dependence structure (i) the entries A i j on and above the diagonal are i.i.d., (i i) for i < j each vector (A i j , A j i) is standard Gaussian with covariance ρ , and independent from the other entries; vector 1 n stands for the n × 1 vector of ones. Parameters α , μ are deterministic and may depend on n. Leveraging on Random Matrix Theory, we analyze this LV system as n → ∞ and study the existence of a positive equilibrium. This question boils down to study the existence of a (componentwise) positive solution to the linear equation: x = r + B x , depending on B 's parameters (α , μ , ρ) , a problem of independent interest in linear algebra. In the case where no positive equilibrium exists, we provide sufficient conditions for the existence of a unique stable equilibrium (with vanishing components), and following Bunin (2017), present a heuristics estimating the number of positive components of the equilibrium and their distribution. The existence of positive equilibria for large Lotka–Volterra systems has been raised in Dougoud et al. (2018), and addressed in various contexts by Bizeul and Najim (2021) and Akjouj and Najim (2022). Such LV systems are widely used in mathematical biology to model populations with interactions, and the existence of a positive equilibrium known as a feasible equilibrium corresponds to the survival of all the species x k within the system. [ABSTRACT FROM AUTHOR]

This paper introduced a numerical method for solving generalized Abel integral (GAI) equations and generalized Abel integro differential (GAID) equations. This method is based upon Touchard polynomials (TPs) approximation. The Touchard polynomials were first presented and the resulting Touchard matrices were utilized to transform the generalized Abel integral and integro differential equations into a system of linear algebraic equations. The results of the presented method were obtained through some examples of the first and second types of equations under study. All results for this method have been compared with those of the presented methods in the literature. [ABSTRACT FROM AUTHOR]

In this work, we introduce the SEE (Sadiq, Emad, Eman) integral transform to find the general solution of the first order complex differential equations with constant coefficients. The usage of SEE integral transform has proven its efficiency in solving a wide range of linear operator equations. [ABSTRACT FROM AUTHOR]

We give new regularity results of solutions for the linear wave equation set in a nonsmooth cylindrical domain. Different types of conditions are imposed on the boundary of the singular domain. Our study is performed in some particular anisotropic Hölder spaces. [ABSTRACT FROM AUTHOR]

The role of Riccati type systems in the plane along with the re-lated linear, second order differential equation is examined. If x and y are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon x alone and are shown to be less important. The systems have both Liouvillian and non{Liouvillian solutions and are easily transformable to sym-metric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations. [ABSTRACT FROM AUTHOR]

We consider a square non linear parametric equations system F(P,X) = 0 which is constituted of n non differential equations in the n unknowns { x 1 , ... , x n } that are the components of X while P = { p 1 , ... , p m } is a set of m parameters that play a role in the definition of the equations F. We assume that P is restricted to lie in a bounded region and we are interested in developing a solver for obtaining all real solutions exactly (a notion that is defined in the paper) for any parameter values within the bounded region. The starting point of the proposed approach is that we assume that a numerical methods has allowed us to determine the real solutions (but not necessarily all of them) for a very limited number of fixed P called the initial solution set. Starting from this set we show that we can create multiple pairs (parameters, solution) and that these pairs may be structured into coherent learning sets that will be used to train multi-layer perceptrons (MLP). The training process is specific: although it still uses a decrease of a loss function its main objective is to maximize the success rate i.e. the number of occurrences, expressed in percentage of number of samples of the training set, for which the Newton scheme, initialized with the MLP prediction, converges toward the expected solution. We then show that for a sufficiently large number of MLPs we may obtain a 100% success rate for all learning sets. The solver is obtained by running a set of local solvers each of which is based on a specific MLP whose prediction may lead to an exact solution of the system. This solver is tested on verification sets i.e. set of samples constituted of parameter values (all different from the samples in the learning set) and all the solutions of the corresponding system. We show that these sets may be automatically generated and that they may also be used in a self-learning process for improving the performance of the solver established from the initial solution set. This approach is illustrated on two engineering problems in robotics and chemistry and it is shown that the solver provides all solutions for any instance of P with a high probability although we cannot guarantee it. The time required to design the final solver is large but the solving time is extremely low so that this approach should be used when the system has to be solved for a sufficient number of occurrences of the P. Furthermore we will show that the computation time required for establishing the solver may be drastically reduced by using a distributed implementation. [ABSTRACT FROM AUTHOR]

In this work we introduce novel stress-only formulations of linear elasticity with special attention to their approximate solution using weighted residual methods. We present four sets of boundary value problems for a pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain. The associated governing equations are derived by modifications and combinations of the Beltrami–Michell equations and the Navier–Cauchy equations. The corresponding variational forms of dimension d ∈ { 2 , 3 } allow to approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in H 1 ⊗ Sym (d) in the framework of functional analysis via the Lax–Milgram theorem, making their finite element implementation using C 0 -continuous elements straightforward. Further, in the finite element setting we provide a treatment for constant and piece-wise constant body forces via distributions. The operators and differential identities in this work are provided in modern tensor notation and rely on exact sequences, making the resulting equations and differential relations directly comprehensible. Finally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and identify viable use-cases of the equations. • New well-posed symmetric variational forms for linear elasticity purely in stresses. • Novel boundary value problems for plane stress and plane strain purely in stresses. • Well-posedness proofs and stabilisation of the planar variational forms. • Treatment of constant and piece-wise constant body forces via distributions. • Numerical convergence benchmarks and spectral analysis of the discretised forms. [ABSTRACT FROM AUTHOR]

We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms. [ABSTRACT FROM AUTHOR]

In this paper, the Volterra-Fredholm integral equation is derived from a linear integrodifferential equation with a fractional order 0 < α < 1 using Riemann--Liouville fractional integral. The existence and uniqueness of the solution are proved using the Picard method. Popular numerical methods; the Toeplitz matrix, and the product Nyström are used in the solution. These methods will prove their effective in solving this type of equation. Two examples are solved using the mentioned methods and the estimation error is calculated. Finally, a comparison between the numerical results is made. [ABSTRACT FROM AUTHOR]

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Electrical resistivity tomography (ERT) is a broadly accepted geophysical method for subsurface investigations. Interpretation of field ERT data usually requires the application of computationally intensive forward modeling and inversion algorithms. For large-scale ERT data, the efficiency of these algorithms depends on the robustness, accuracy, and scalability on high-performance computing resources. In this regard, we present a robust and highly scalable implementation of forward modeling and inversion algorithms for ERT data. The implementation is publicly available and developed within the framework of PFLOTRAN, an open-source, state-of-the-art massively parallel subsurface flow and transport simulation code. The forward modeling is based on a finite volume discretization of the governing differential equations, and the inversion uses a Gauss-Newton optimization scheme. To evaluate the accuracy of the forward modeling, two examples are first presented by considering layered (1D) and 3D earth conductivity models. The computed numerical results show good agreement with the analytical solutions for the layered earth model and results from a well-established code for the 3D model. Inversion of ERT data, simulated for a 3D model, is then performed to demonstrate the inversion capability by recovering the conductivity of the model. To demonstrate the parallel performance of PFLOTRAN's ERT process model and inversion capabilities, large-scale scalability tests are performed by using up to 131,072 processes on a leadership class supercomputer. These tests are performed for the two most computationally intensive steps of the ERT inversion: forward modeling and Jacobian computation. For the forward modeling, we consider models with up to 122 million degrees of freedom in the resulting system of linear equations, and demonstrate that the code exhibits almost linear scalability on up to 8,192 cores. On the other hand, the code shows perfectly linear scalability for the Jacobian computation, mainly because all computations are fairly evenly distributed over each core with no parallel communication. [ABSTRACT FROM AUTHOR]

This work focuses to solve any order of scalar differential equation involved in analog circuit representation. These types of mathematical representations have many applications in analysis and processing such as noise, filter, audio, RLC distributed interconnection (nodes) and transmission lines. Such systems are represented with scalar type differential equations and use numerical method to find a solution. One of the most successful methods is the fourth-order Runge–Kutta. This study introduced a multiplicative version of Runge–Kutta (MRK4) method. The performance analysis of the MRK4 is examined based on the error analysis method. The MRK4 method is applied to solve equations representing the linear and the nonlinear type systems. Results indicate the MRK4 to be superior with respect to the RK4 method. [ABSTRACT FROM AUTHOR]