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In this article, we consider an inverse problem related to the damped generalized regularized long wave equation with noisy data. We study numerically this problem by applying the quartic B-spline ...

In the present paper, we study one-dimensional inverse source problem of identifying a time-dependent source function from the over-specification condition. The unknown source term is recovered by solving an operator integral equation of the first kind. Due to the ill-posedness of this operator integral equation, the Tikhonov regularization approach of the 1st order is applied in order to acquire a stable approximation. The stable solution is defined by minimization of the Tikhonov functional. The value of the regularization parameter in this method is obtained as a function of error in input data. The finite element method is applied for numerical simulation based on obtained results. The illustrative example is presented to show the accuracy and applicability of the proposed method.

In this paper, a numerical method consists of combining Haar wavelet method and Tikhonov regularization method to determine unknown boundary condition and unknown nonlinear source term for the generalized time-delayed Burgers-Fisher equation using noisy data is presented. A stable numerical solution is determined for the problem. We also show that the rate of convergence of the method is as exponential $ \Bigl(O\left(\frac{1}{2^{J+1}}\right)\Bigr) $, where $ J $ is maximal level of resolution of wavelet. Some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.

In this paper, two numerical techniques are presented to solve the nonlinear inverse generalized Benjamin–Bona–Mahony–Burgers equation using noisy data. These two methods are the quartic B-spline and Haar wavelet methods combined with the Tikhonov regularization method. We show that the convergence rate of the proposed methods is $$\textit{O}(k^2+h^3)$$ and $$\textit{O}\left( \frac{1}{M}\right) $$ for the quartic B-spline and Haar wavelet method, respectively. In fact, this work considers a comparative study between quartic B-spline and Haar wavelet methods to solve some nonlinear inverse problems. Results show that an excellent estimation of the unknown functions of the nonlinear inverse problem has been obtained.

In order to solve the nonlinear time fractional inverse parabolic equations, two methods namely, combination of the finite difference method (FDM) and cubic B-spline collocation method are proposed...

In this paper, a numerical method is proposed for the numerical solution of the coupled nonlinear reaction–diffusion equations with suitable initial and boundary conditions by using the Haa...

In this paper, we propose a meshless method using radial basis functions (RBFs) method based on finite-difference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. Indeed, the aim of this paper is to show that the meshless method based on the radial basis functions approach is also suitable for the treatment of the inverse nonlinear partial differential equations. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. We prove that the time discrete scheme is stable using the energy method. The effectiveness of the proposed method is illustrated by numerical examples.

In this paper, based on the cubic B-spline finite element (CBSFE) and the radial basis functions (RBFs) methods, the inverse problems of finding the nonlinear source term for system of reaction–diffusion equations are studied. The approach of the proposed methods are to approximate unknown coefficients by a polynomial function whose coefficients are determined from the solution of minimization problem based on the overspecified data. In fact, this work considers a comparative study between the cubic B-spline finite element method and radial basis functions method. The stability and convergence analysis for these problems are investigated and some examples are given to illustrate the results.

In this paper, a numerical method is proposed for the numerical solution of a linear system with suitable initial and boundary conditions using the cubic B-spline collocation scheme to determine th...

In this paper, a spectral Tau method based on Legendre Wavelet basis is proposed. For this purpose we present a stable operational Tau method based on Legendre Wavelet basis. This method provides an efficient approximate solution for weakly singular Volterra integral equations by using reduced set of matrix operations. An error estimation of the Tau method is also introduced. Finally we demonstrate the validity and applicability of the method by numerical examples.

Text Define χ ( n ) recursively by χ ( 1 ) = 1 and χ ( n ) = ϕ ( n ) + χ ( n / q ) for all integers n > 1 , where q is the least prime factor of n, and where ϕ is the Euler totient function. We show that χ ( n ) = ϕ ( d ) ( χ ( l ) − 1 ) + χ ( d ) , where n = d l and the prime factors of d are greater than the prime factors of l. We also show χ ( n m ) ≤ χ ( n ) χ ( m ) when n and m are coprime numbers. As an application, we show that for all primes p ≥ 11 , χ ( p 2 − p ) > χ ( p 2 − 1 ) . We discuss the interpretation of χ as the clique number of the power graph of a finite cyclic group and the significance of the inequality in this context. Video For a video summary of this paper, please visit https://youtu.be/p8finzAEJps .

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and derivatives, which produce an ill-posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

In this paper a numerical approach combining the least squares method and the genetic algorithm (sequential and multi-core parallelization approach) is proposed for the determination of temperature in an inverse heat conduction problem (IHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.7 GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.7 GHz for each core.

In this paper, we study the ADMB–KdV equation. We show that the associated initial value problem is locally well posed in Sobolev spaces \(H^{s}(\mathbb{R}^{2})\) for s>−1/2. We also prove that our result is sharp in some sense.

We show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph. Contains corrections to published version., Comment: second set of corrections to technical lemmas, no changes to main result

In this paper we study the solitary waves of the Rosenau-RLW equation. By using some trigonometric function methods, a family of stable solitary wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed, for what should be an equation relevant to modeling in a number of fields.

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example.

This paper studies the ADM-BKdV equation. The associated initial value problem will be proved to be locally well-posed in anisotropic Sobolev spaces \(H^{s_1,s_2}\left( \mathbb R ^2\right) \) for \(s_1>-3/2,\,s_2>-1/2\) and \(2s_1+4s_2>-1\).

In this paper a numerical method consists of combining Haar basis method and Tikhonov regularization method for solving some ill-posed inverse problems using noisy data is presented. By using a sensor located at a point inside the body and measuring the u(x, t) at a point x = a, 0 < a < 1, and applying Haar basis method to the inverse problem, we determine a stable numerical solution to this problem. Results show that an excellent estimation on the unknown functions of the inverse problem can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f , g , and for a properly specified class A of admissible coefficients, there exists at least one a ∈ A such that ( a , u ) is a solution of the corresponding inverse problem.

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved.

A numerical approach combining the use of the finite difference method with the solution of ordinary differential equation (ODE) is proposed for the determination of boundary conditions in an inverse parabolic problem. The least-squares method is adopted to modify the solution. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

This study is intended to provide a new numerical algorithm for solving a one-dimensional inverse nonlinear parabolic problem. At the beginning of the study, Taylor’s series expansion is employed to linearize nonlinear terms and then finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate unknown functions. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown coefficient in the present study, thus, it is classified as the function estimation. Results show that an excellent estimation on the unknown functions of the inverse problem which can be obtained within a couple of minutes CPU time at Pentium IV-2.4 GHz PC.

In this paper, we consider an inverse heat conduction problem (IHCP). The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a linear transformation, the ill-posed IHCP becomes a Cauchy problem. This problem will be solved by applying Legendre polynomials. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

In this paper, a numerical solution of an inverse non-dimensional heat conduction problem will be considered. By using a sensor located at a point inside the body and measuring the temperature at a point x = x 1 , 0 x 1

A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite group of order $n$, and let $\text {C}_n$ be the cyclic group of order $n$. For $g\in G$, let ${\mathrm{o}}(g)$ denote the order of $g$. Let $\phi $ denote the Euler totient function. We show that $\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$, with equality if and only if $G$ is isomorphic to $\text {C}_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of bidirectional edges in its directed power graph.