Certain Types Of Differential Equations Research Articles

Application of Laplace Transform in Solving Linear Differential Equations with Constant Coefficients

In recent years, the interest in using Laplace transforms as a useful method to solve certain types of differential equations and integral equations has grown significantly. In addition, the applications of Laplace transform are closely related to some important parts of pure mathematics. Laplace transform is one of the methods for solving differential equations. This method is especially useful for solving inhomogeneous differential equations with constant coefficients and it has advantages compared to other methods of solving differential equations. Linear differential equations with constant coefficients are among the equations that can be solved using the Laplace transform. Because the transformation Laplace is one of the transformations that easily converts exponential functions, trigonometric functions, and logarithmic functions into algebraic functions. Therefore, it is considered a better method for solving linear differential equations with constant coefficients.

Transcendental meromorphic solutions of certain types of differential equations

In this paper, we analyze the transcendental meromorphic solutions of nonlinear complex differential equation of the formfn+P(z,f)=p1eα1zk+p2eα2zk, where n≥2 and k≥1 are integers, f is a meromorphic function, P(z,f) is a differential polynomial in f of degree at most n−1 with rational functions of f as the coefficients, α1,α2,p1,p2 are nonzero constants. We also present a number of examples showing the accuracy of our results.

Value distribution of meromorphic functions satisfying certain type of differential equations

Value distribution of meromorphic functions satisfying certain type of differential equations

Network Completion Using Dynamic Programming and Least-Squares Fitting

We consider the problem of network completion, which is to make the minimum amount of modifications to a given network so that the resulting network is most consistent with the observed data. We employ here a certain type of differential equations as gene regulation rules in a genetic network, gene expression time series data as observed data, and deletions and additions of edges as basic modification operations. In addition, we assume that the numbers of deleted and added edges are specified. For this problem, we present a novel method using dynamic programming and least-squares fitting and show that it outputs a network with the minimum sum squared error in polynomial time if the maximum indegree of the network is bounded by a constant. We also perform computational experiments using both artificially generated and real gene expression time series data.

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H c (x) on the collocation points is used for the approximation of singular source terms δ(x-c) or δ (n)(x-c) without any regularization. The direct projection of H c (x) yields highly oscillatory approximations of δ(x-c) and δ (n)(x-c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided.

Entire solutions of certain type of differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, we solve the transcendental entire solutions of the following type of nonlinear differential equations in the complex plane: f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z , where p 1 and p 2 are two small functions of e z , and α 1 , α 2 are two nonzero constants with some additional conditions, and P ( f ) denotes a differential polynomial in f and its derivatives (with small functions of f as the coefficients) of degree no greater than n − 1 .

Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation

Ulam’s problem for approximate homomorphisms and its application to certain types of differential equations was first investigated by Alsina and Ger. They proved in [C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380] that if a differentiable function f : I → R satisfies the differential inequality ∣ y′( t) − y( t)∣ ⩽ ε, where I is an open subinterval of R , then there exists a solution f 0 : I → R of the equation y′( t) = y( t) such that ∣ f( t) − f 0( t)∣ ⩽ 3 ε for any t ∈ I. In this paper, we investigate the Ulam’s problem concerning the Bernoulli’s differential equation of the form y( t) − α y′( t) + g( t) y( t) 1− α + h( t) = 0.

Hamilton's Principle for a Set of Nonlinear Heat Conduction

We calculate the Lagrangian for certain type of differential equations of nonlinear heat conduction, applying potential function method introduced previously. On the other hand, we point out that these kind of nonlinear parabolic differential equations describe Markovian processes in a new phase space.

Computational modelling with functional differential equations

Mathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in ...

Computational modelling with functional differential equations: Identification, selection, and sensitivity

Mathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in particular biological, phenomena. We present some of the principles underlying the choice of a methodology (based on observational data) for the computational identification of, and discrimination between, quantitatively consistent models, using scientifically meaningful parameters. We propose that a computational approach is essential for obtaining meaningful models. For example, it permits the choice of realistic models incorporating a time-lag which is entirely natural from the scientific perspective. The time-lag is a feature that can permit a close reconciliation between models incorporating computed parameter values and observations. Exploiting the link between information theory, maximum likelihood, and weighted least squares, and with distributional assumptions on the data errors, we may construct an appropriate objective function to be minimized computationally. The minimizer is sought over a set of parameters (which may include the time-lag) that define the model. Each evaluation of the objective function requires the computational solution of the parametrized equations defining the model. To select a parametrized model, from amongst a family or hierarchy of possible best-fit models, we are able to employ certain indicators based on information-theoretic criteria. We can evaluate confidence intervals for the parameters, and a sensitivity analysis provides an expression for an information matrix, and feedback on the covariances of the parameters in relation to the best fit. This gives a firm basis for any simplification of the model (e.g., by omitting a parameter).

ASYMPTOTIC EQUILIBRIUM FOR CERTAIN TYPE OF DIFFERENTIAL EQUATIONS WITH MAXIMUM

In this work we obtain asymptotic representations for the solutions of certain type of differential equations with maximum. We deduce the asymptotic equilibrium for this class of differential equations.

An improved method for computing a discrete Hankel transform

A new and simple algorithm for computing a discrete Hankel transform, which does not rely on the fast Fourier transform is described. In the solution of certain types of differential equations this algorithm can provide a major improvement in speed and accuracy over previously described methods. An application to a particular form of tile transport equation relevant to a class of problems in electron scattering is given.

Problem of justification of the averaging method for a certain type of differential equations with deviating arguments

Problem of justification of the averaging method for a certain type of differential equations with deviating arguments

Steady-state solutions of certain types of differential equations by piecewise linearization

Two methods are presented for finding steady-state solutions of differential equations of any order governing certain systems that are acted upon by a harmonic force and have one nonlinear element with hysteresis represented by piecewise linearization. Both methods need the solution of a set of linear algebraic equations. In the first method, the unknowns are the Fourier coefficients of the steady-state solution, while in the second method, the unknowns are the values of the derivatives of the steady-state solution at a break point of the piecewise linearized characteristic. In both methods, the unknowns have to be calculated for different values of the time angle at the break point, yielding different corresponding values of the amplitude of the forcing term. The required solution is that consistent with the given amplitude of this forcing term. In the first method, the parameters involved in the multiple-input describing functions of the nonlinear element are unified by normalization. Comparison of the two methods is given, and the advantages of the piecewise linearization of characteristics is discussed.

Some Exercises in Laplace Transform Integrals

If 𝜙(t) is a given function of the real variable t then the integral (in which p is constant) if convergent, is defined to be the Laplace transform of 𝜙(t). The use of the integral in determining the solutions of certain types of differential equations is now well known; see, for example, Carslaw and Jaeger, Operational methods in applied mathematics (Oxford University Press).