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A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss. The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions. Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.
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276Seven common errors in finding exact solutions of nonlinear differential equations
Seven common errors in finding exact solutions of nonlinear differential equations
12Solution of nonlinear partial differential equations from base equations
Solution of nonlinear partial differential equations from base equations
11A new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients
The main purpose of this paper is to present a new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients using the fractional residual power series method (FRPSM). The fractional derivative is considered in the Caputo sense. This method is based on the generalized Taylor series formula and residual error function. Unlike other analytical methods, FRPSM has a special advantage, that it solves the nonlinear problems without using linearization, discretization, perturbation or any other restrictions. By numerical examples, it is shown that the FRPSM is a simple, effective, and powerful method for finding approximate analytical solutions of nonlinear fractional partial differential equations.
Read more34Global Error Minimization method for solving strongly nonlinear oscillator differential equations
Global Error Minimization method for solving strongly nonlinear oscillator differential equations
5Approximate of solution of a fourth order ordinary differential equations via tenth step block method
This paper carries a different approach of collection and interpolation to develop a tenth block method for the numerical solution of linear or nonlinear ordinary differential equations of fourth order with initial conditions. The method has been implemented at the selected mesh points to generate a direct tenth block method through Taylor series. Some critical properties of this method such as zero stability, order of the method, and convergence have been analysed. Two numerical tests have taken to make a comparison of the approximate results with exact as well as results of other authors.
Read more31An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics
The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.
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11A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations
This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques.
Read more87The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations
The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations
26Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method
Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method
Read more8Existence results for time scale boundary value problem
Existence results for time scale boundary value problem
Doubly periodic meromorphic solutions of autonomous nonlinear differential equations
The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.
Read moreDoubly Periodic Meromorphic Solutions of Autonomous Nonlinear Differential Equations
The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.
Read more57The Numerical Solution of Integral Equations on the Half-Line
The Numerical Solution of Integral Equations on the Half-Line
27Comparison of solutions of linear and non-linear shallow water wave equations using homotopy perturbation method
PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.
Read more13On the Summability of Formal Solutions of Nonlinear Partial Differential Equations with Shrinkings
We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.
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