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Second-order linear hyperbolic equations are solved by using a new three level method based on nonpolynomial spline in the space direction and Taylor expansion in the time direction. Numerical results reveal that three level method based on non-polynomial spline is implemented and effective

A new technique using fractal non-polynomial spline is formulated for approximating the solution of certain fourth-order two-point boundary value problems arising in plate deflection theory. Numerical techniques of second, fourth and almost sixth order are derived which provide five-diagonal linear systems. Convergence analysis of sixth-order method is discussed. Numerical examples are presented to prove the supremacy of developed technique over fractal quintic spline and some other spline techniques.

Quartic non-polynomial spline method is presented to solve the singularly perturbed differential-difference equation containing two parameters. The considered equation is transformed into an asymptotical equivalent differential equation, and the derivatives are replaced finite difference approximation using the quartic non-polynomial spline method. The convergence analysis of the method has been established. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Furthermore, the present method gives a more accurate solution than some existing methods reported in the literature.

This paper describes the exceptionally precise results of 6 th -order and 8 th -order nonlinear boundary-value problems(BVPs). Cubic-Nonpolynomial spline(CNPS) and Cubic-polynomial spline(CNPS) are utilized to solve such types of BVPs. We develop the class of numerical techniques for a particular selection of the factors that are associated with nonpolynomial and polynomial splines. Using the developed class of numerical techniques, the problem is reduced to a new system that consists of 2 nd -order BVPs only. The end conditions associated with the BVPs are determined. For each problem, the results obtained by CNPS and CPS is compared with the exact solution. The absolute error(AE) for every iteration is calculated. To show that the suitable responses established by using CNPS and CPS have a higher level of preciseness, the absolute errors of the CNPS and CPS have been compared with different techniques such as DTM, ADM, Parametric septic splines, Variational-iteration method(VIM), Daftardar Jafari strategy, MDM, Cubic B-Spline, Homotopy method(HM), Quintic and Sextic B-spline and observed to be more accurate. Graphs that describe the graphical comparison of CNPS and CPS at n = 10 are also included in this paper.

The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous p(x)- Laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.

The application of the local polynomial and non-polynomial to the construction of methods for numerically solving the heat conduction problem is discussed. The non-polynomial splines are used here to approximate the partial derivatives. Formulas for numerical differentiation based on the application of the nonpolynomial splines of the fourth order of approximation are constructed. Particular attention is paid to polynomial, trigonometric, exponential, polynomial-trigonometric and polynomial-exponential splines. This approach allows us to construct explicit and implicit difference schemes. The main focus of the paper is on implicit difference scheme. New approximations with splines of the Lagrange and Hermite type with new properties are obtained. These approximations take into account the first and second derivatives of the function being approximated. Numerical examples are given.

Parabolic partial differential equation having a great impact on our scientific, engineering and technology. Enormous research have been conducted for the solution of parabolic PDEs. . In this research work, we introduced a novel technique for the numerical solution of fourth order PDEs. The novel technique is based upon the polynomial cubic cutting method (PCSM) was used with Adomian breakdown technique (ADM).The constraint for the alternative variables was decomposed by Edomian decomposition for the successive approximation. A numerical test problem of parabolic PDEs solved by purposed technique

This study introduces a non-polynomial septic spline method for solving singularly perturbed two point boundary value problems of order three. First, the given interval is discretized. Then, the spline coefficients are derived and the consistency relation is obtained by using continuity of second, fourth and fifth derivatives. Further, the obtained fifteen different systems of equations are reduced to a system of equations and boundary equations are developed in order to equate a system of linear equations. The convergence analysis of the obtained hepta-diagonal scheme is investigated. To validate the applicability of the method, two model examples are considered for different values of perturbation parameter $\varepsilon $ and different mesh size h. The proposed method approximates the exact solution very well when $\varepsilon \ll h $. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

Non-polynomial quadratic spline method based on off-step points is used to develop a numerical algorithm for obtaining an approximate solution of higher even order boundary value problems. For the employment of the method, decomposition procedure is used. Higher order boundary value problems are reduced into the corresponding system of second order boundary value problems. Convergence analysis of the method is also discussed. Seven numerical examples are given to illustrate the applicability and efficiency of new method. It is also shown that the new method gives approximations, which are better than those produced by other existing fourth order methods except higher degree splines.

In this paper, we propose a new numerical scheme for two-dimensional fractional sub-diffusion problems using non-polynomial spline. The solvability, stability and convergence of the proposed method are established using the well known discrete energy methodology. It is shown that the spatial convergence order is at least 4.5 which improves the best result achieved to date. We also carry out simulation to demonstrate the accuracy and efficiency of the proposed scheme and to compare with other methods.

This article presents the numerical solution of a system of linear fourth-order boundary value problems using a different amalgamation of non-polynomial splines. A novel approach was developed usin...

In this paper, different variants of processing of number flows using Lagrange and Hermite non-polynomial splines are studied. The splines are constructed from approximate relations including a generating vector function with components of different character, including non-polynomial. Approximations by first-order Lagrange and third-order Hermite splines are considered. The efficiency of the approximations constructed is demonstrated on the examples of flows of the values of a function and flows of the values of a function and its derivative. The advantages of the splines considered are the simplicity of construction, maximum smoothness, interpolation and approximation properties, and the accuracy on a priori given functions (on the components of the generating vector function).

In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining n...

We construct certain non-polynomial entire solutions to $$\sigma _k$$ equations on $${\mathbb {R}}^n$$ .

An algorithm for the identification of non-polynomial calibration dependences, which can be reduced to linear dependences by change of variables, has been developed. The parameters of linear dependences are found by the least-squares method. An algorithm for measurement uncertainty evaluation using the identified dependence is given, accounting for the instrumental measurement uncertainties of the responses of the calibrated measuring instrument and the values of the calibration points.

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and $(2-\gamma )$ ( 2 − γ ) th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction a x 2 + b x 2 /(1 + c x 2 ) ( a > 0, c > 0) are given by the confluent Heun functions H c ( α , β , γ , δ , η ; z ). The minimum value of the potential well is calculated as V min ( x ) = − ( a + | b | − 2 a | b | ) / c at x = ± [ ( | b | / a − 1 ) / c ] 1 / 2 (| b | > a ) for the double-well case ( b a , b , c and show that they are pulled back to the origin when the potential parameter b increases for given values of a and c . However, we find that the wave peaks are concave to the origin as the parameter | b | is increased.

This paper studies the limit cycles produced by small perturbations of certain planar Hamiltonian systems. The limit cycles under consideration correspond to critical levels of the Hamiltonian, that is they are located in a small vicinity of a separatrix contour or a critical point. Two most interesting facts in the paper are that the Hamiltonian function is not a polynomial and that the system under consideration comes from a model of oscillator with a pair of irrational nonlinearities, which implies the transition from smooth to discontinuous dynamics. This model has been proposed recently by Han et al. in a paper published in 2012.

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

In this paper, we propose a cubic non-polynomial spline method to solve the time-fractional nonlinear Schrodinger equation. The method is based on applying the $L_{1}$ formula to approximate the Caputo fractional derivative and employing the cubic non-polynomial spline functions to approximate the spatial derivative. By considering suitable relevant parameters, the scheme of order $O(\tau^{2-\alpha }+h^{4})$ has been obtained. The unconditional stability of the method is analyzed by the Fourier analysis. Numerical experiments are given to illustrate the effectiveness and accuracy of the proposed method.

In this paper, we tackle the numerical treatment of a fourth-order fractional diffusion-wave problem. By using parametric quintic spline in the spatial dimension and an approximation of Caputo derivatives at half-points, we propose a numerical scheme and rigorously prove its solvability, convergence and stability in maximum norm. It is shown that the theoretical convergence order improves those of earlier work. To confirm, simulation is carried out to demonstrate the numerical efficiency of the proposed scheme as well as the better performance over other methods.

In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.

This comment paper aims to show that the paper (Adhikari and Singh [1]) contains a serious error. In the commented paper, the displacements are incorrect—two of the shape functions they are based on are erroneous, and they do not properly account for the same degree. The correct forms of the shape functions are explained more deeply here.

The application of the Isogeometric Analysis (IgA) paradigm to Symmetric Galerkin Boundary Element Method (SGBEM) is investigated. In order to obtain a very flexible approach, the study is here developed by using non-polynomial spline functions to represent both the domain boundary and the approximate solution. The numerical comparison between IGA-SGBEM and both curvilinear and standard SGBEM approaches shows the general capability of the presented method to produce accurate approximate solutions with less degrees of freedom.

In this paper, non-polynomial spline method for solving one dimensional nonlinear Benjamin-Bona-Mahony-Burgers equation is presented. Stability analysis of the present method is analyzed by means of Von Neumann process and the method is proven to be unconditionally stable. Truncation error of the proposed method is also discussed. Few numerical evidences are given to prove the validation of the proposed method.

In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.

We show that two natural extensions of the real Casas-Alvero conjecture in the non-polynomial setting do not hold.

A numerical scheme is proposed using a non polynomial spline to solve the differential-difference equations having layer behaviour, with delay as well advanced terms. The retarded terms are handled by using Taylor’s series, subsequently the given problem is substituted by an equivalent second order singular perturbation problem. A finite difference scheme using non polynomial spline is derived and it is applied to the singular perturbation problem using non standard differences of the first derivatives. Tridiagonal algorithm is used to solve the resulting system. The method is exemplified on numerical examples with various values of perturbation, delay and advance parameters. Maximum absolute errors are computed and tabulated to support the method. Numerical solutions are pictured in graphs and the effects of small shifts on the boundary layer region has been investigated. Also, the convergence of the proposed method has also been established.

Two-point boundary value problems are commonly used as a numerical test in developing an efficient numerical method. Several researchers studied the application of a cubic non-polynomial spline method to solve the two-point boundary value problems. A preliminary study found that a cubic non-polynomial spline method is better than a standard finite difference method in terms of the accuracy of the solution. Therefore, this paper aims to examine the performance of a cubic non-polynomial spline method through the combination with the full-, half-, and quarter-sweep iterations. The performance was evaluated in terms of the number of iterations, the execution time and the maximum absolute error by varying the iterations from full-, half- to quarter-sweep. A successive over-relaxation iterative method was implemented to solve the large and sparse linear system. The numerical result showed that the newly derived QSSOR method, based on a cubic non-polynomial spline, performed better than the tested FSSOR and HSSOR methods.

In this work, the fifth order non-polynomial spline functions are used to solve linear Volterra integral equations with weakly singular kernel. Numerical example is presented to illustrate the applications of this method and to compare the computed results with other numerical methods. we take integral equation when it has the kernel and it is non-linear. The results are the approximations that depended on Volterra integral equation. Our work calculated the results for the Volterra integral equation, which are approximate that the error is found.

This paper presents a free vibration analysis of plates made of functionally graded materials and resting on twolayer elastic foundations by proposing a non-polynomial four variable refined plate theory. Undetermined integral terms are introduced in the proposed displacement field and unlike the conventional higher shear deformation theory (HSDT), the present one contains only four unknowns. Equations of motion are derived via the Hamilton\'s principles and solved using Navier\'s procedure. Accuracy of the present theory is demonstrated by comparing the results of numerical examples with the ones available in literature.

In this paper, we propose a quintic non-polynomial spline approach coupled with $L1$ formula for the numerical treatment of time-fractional nonlinear Schrodinger equation. The unconditional stability is proved by Fourier method for $0 , where $\gamma$ is the order of the Caputo time-fractional derivative. The efficiency of the proposed numerical scheme is illustrated by numerical experiments, where the simulation results indicate better performance over previous work in the literature.

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature. Published version

In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.

A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of logarithmically singular integrals. The interpolation and integration errors are also analyzed. Numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature., 13 pages, 2 figures, 8 tables, accepted by Journal of Computational and Applied Mathematics

This paper presents a free vibration analysis of functionally graded plates resting on elastic foundation by using a generalized quasi-3D hybrid-type higher order shear deformation theory (HSDT). The displacement field is modeled based on a hybrid-type (hyperbolic and sinusoidal) quasi-3D HSDT with six unknowns in which the stretching effect is taken into account. The elastic foundation follows the Pasternak (two-parameter) mathematical model. The governing equations are obtained through the Hamilton’s principle. These equations are then solved via Navier-type, closed form solutions. The fundamental frequencies are found by solving the eigenvalue problem. The performance of the present theory is demonstrated by comparing results with the 3D exact solution and other closed form solutions available in literature.

This paper presents Best Theory Diagrams (BTDs) constructed from non-polynomial terms to identify best shell theories for the free vibration analysis of laminated and sandwich shell. The shell theories have been constructed using Axiomatic/Asymptotic Method (AAM). The refined models are implemented following the compactness of a unified formulation developed. The governing equations are derived from the Hamilton’s Principle. Navier-Type solution technique is used for solving the eigenvalue problem of simply supported shell. The BTDs use 3D equilibrium solutions as a reference. The BTDs built from non-polynomial functions are compared with Maclaurin expansions. The results are compared with Layerwise solutions. Cylindrical and spherical shells with different layer-configurations are investigated. The results demonstrate that the shell models obtained from the BTD using non-polynomial terms can improve the accuracy obtained from Maclaurin expansion for a given order of expansion of a displacement field.

We present a review on Lagrangian models admitting spherically symmetric regular black holes, and cosmological bounce solutions. Non-linear electrodynamics, non-polynomial gravity, and fluid approaches are explained in details. They consist respectively in a gauge invariant generalization of the Maxwell Lagrangian, in modifications of the Einstein-Hilbert action via non-polynomial curvature invariants, and finally in the reconstruction of density profiles able to cure the central singularity of black holes. The non-polynomial gravity curvature invariants have the special property to be second order and polynomial in the metric field, in spherically symmetric spacetimes. Along the way, other models and results are discussed, and some general properties that regular black holes should satisfy are mentioned. A covariant Sakharov criterion for the absence of singularities in dynamical spherically symmetric spacetimes is also proposed and checked for some examples of such regular metric fields., 39 pages; submitted and accepted for publication in Int. J. Mod. Phys. D; references added

In this paper, we propose a new high accuracy method based on non-polynomial spline for the numerical solution of two-dimensional elliptic partial di erential equations. Using a non-polynomial spline approximation in the xdirection and central di erence in the y- direction, we obtain a new nine-point compact finite di erence formulation. A four point group explicit (GE) iterative scheme with an acceleration tool is then applied to the obtained system. The formulation procedure is presented in detail. The e ciency of the proposed method is then illustrated by some test problems. The numerical results are found to be in good agreement with the exact solutions

Sudoku is one of the popular puzzle games in many countries. How to solve this game is being the Non Polynomial Completeness problem. Because of that, a lot of research has been done to find the most effective algorithm completing Sudoku. This study presents the evaluation of algorithm in completing the Sudoku 2 x 2 and 3 x 3. The most common algorithm is brute force and backtracking. There is also simulated annealing algorithm. The evaluation is done by comparing the step of each algorithm to solve the Sudoku and also their running time. The result of this evaluation shows the differences of each algorithm characteristic in solving this Sudoku problem. By comparing the result of this study, the most effective algorithm can be chosen. Based on the test, concluded that the backtracking algorithm is more effective than the simulated annealing algorithm which the value reach 50% .

The question of interpolation of a function of one variable with large gradients in the boundary layer region is investigated. The problem is that applying of polynomial interpolation formulas on a uniform grid to functions with large gradients can lead to unacceptable errors. We study the interpolation formulas with an arbitrarily number of interpolation nodes which are exact on the singular component. This component is responsible for the main growth of the function in the boundary layer and can be found based on asymptotic expansions. It is proved that error estimates don’t depend on the singular component and its derivatives. In the case of an exponential boundary layer these estimates don’t depend on a small parameter.

Bäcklund transformation (BT) is an intrinsic property of nonlinear integrable system. Generally, it is an interesting and challenging work to investigate BT of a nonlinear system, especially for the non-polynomial form. In this paper, we introduce an analytic method for constructing BTs of the generalized nonlinear wave equations (NLWEs) of the form utt=auxx+h(u). The BTs with arbitrary parameters are provided explicitly, so the integrability of the equation is verified accordingly. Then, the nonlinear superposition formulas (NSFs) of the NLWEs are given in terms of such BTs, and the infinite number of exact explicit solutions to the equations are obtained based on the BTs and the NSFs. Furthermore, the BTs of the other types of NLWEs of the form uxt=h(u) can be provided directly by the variable transformation method.