Integration across discontinuities in ordinary differential equations using power series

The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales $R_k$ in the power term $(t/R_k)^k$ is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

Domain-partition power series in vibration analysis of variable-cross-section rods

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean’s stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamical systems in various scientific fields. However, solving ODEs analytically can be challenging, and numerical methods can be computationally expensive or inaccurate. In recent years, deep neural networks (DNNs) have emerged as a promising alternative for solving ODEs. This paper focuses on the utilization of DNNs to solve first and second-order ODEs. Existing works in this field are reviewed, with an emphasis on identifying their limitations. Subsequently, an approach is proposed to overcome these limitations. The proposed method utilizes a combination of a deep neural network and various activation functions to achieve accurate and efficient solutions for ordinary differential equations (ODEs). Deep neural networks (DNNs) have shown significant potential in solving ordinary differential equations (ODEs) due to their ability to approximate complex functions and their capacity to learn from data. Traditional numerical methods for solving ODEs have some limitations such as requiring large amounts of computation, limited accuracy, and difficulty in handling high-dimensional systems. The aim is to overcome these limitations by leveraging the capabilities of DNNs, that have the potential to learn and generalize from data, provide more accurate solutions, requiring fewer computational resources, and being able to handle high-dimensional systems. This paper aims to explore the use of DNNs in solving 1st and 2nd order differential equations, and proposes a novel approach for improving the accuracy and efficiency of the solution process.

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

We address finding solutions y ∈ ʗ2 (ℝ+) of the special (linear) ordinary differential equation xy''(x) + (ax2 + b) y' (x) + (cx + d)y(x) = 0 for all x ∈ ℝ+, where a, b, c, d ∈ ℝ are constant parameters. This will be achieved in three special cases via separation and a power series method which is specified using difference equation techniques. Moreover, we will prove that our solutions are square integrable in a weighted sense—the weight function being similar to the Gaussian bell in the scenario of Hermite polynomials. Finally, we will discuss the physical relevance of our results, as the differential equation is also related to basic problems in quantum mechanics.

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point 0 t = and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

In this work, the Lie group method is used to perform the similarity reduction and to obtain the exact solutions of the $$(2+1)$$ -dimensional KdV-Burgers equation. Similarity transformation method reduces $$(2+1)$$ -dimensional KdV-Burgers equation into $$(1+1)$$ -dimensional PDEs, later it reduces these PDEs into various ordinary differential equations and helps to find exact solutions of $$(2+1)$$ -dimensional KdV-Burgers equation. With the help of reduced equations, we have obtained the exact explicit solutions. Moreover, later by power series method, the exact analytic solutions of the KdV-Burgers equation are obtained.

In this work, we study the generalized nonlinear Schrodinger equation having time dependent variable coefficients by using Lie group method. We find the Lie point symmetries admitted by the Schrodinger equation for arbitrary variable coefficients and from the obtained symmetries, we constructed the similarity reductions. As the further symmetry analysis of the reduced ordinary differential equations was not possible due to lack of symmetries, we, therefore, solve one of the reduced nonlinear ordinary differential equation for completeness by using power series method and presented the exact series solution. The convergence of the series solutions is also established.

Automatic differentiation with the use of computer allows one to evaluate the Taylor series of expansion of a function acurately and easily. After showing automatic differentiation of a class of functions, the power series of solutions of ordinary differential equation (ODE) and the system of ODE are explained.