Higher-order Equations Research Articles

Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions

In this paper, we considered an inverse problem of recovering the space-dependent source coefficient in the third-order pseudo-parabolic equation from final over-determination condition. This inverse problem appears extensively in the modelling of various phenomena in physics such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, etc. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the final input may cause relatively significant errors in the output source term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown force term. The third-order pseudo-parabolic problem is discretized using the Cubic B-spline (CB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Both perturbed data and analytical solutions are inverted. Numerical outcomes are reported and discussed. The computational efficiency of the method is investigated by small values of CPU time. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

On the Use of Elzaki Decomposition Method for Solving Higher-Order Integro-Differential Equations

In this article, we implemented the Elzaki decomposition technique (EDM) to solve Volterra–Fredholm integro-differential equations of higher-order. Illustrations are used to test the technique’s accuracy and validity. Comparison among the acquired consequences by EDM and actual solutions have proven the power and accuracy of this technique. This technique is dependable and able to supply analytic remedies for solving such equations.

Coherence and Stochastic Resonances in a Noisy van der Pol-Type Circadian Pacemaker Model Driven by Light

Daylight plays a major role in the wake/sleep cycle in humans. Indeed, the wake/sleep system stems from biological systems that follow a circadian rhythm determined by the light/dark alternation. The oscillations can be modeled by the higher-order nonlinear van der Pol-type equation driven by a term that mimics the light cycle. In this work, noise in the illumination is introduced to investigate its effect on the human circadian cycle. It is found that the presence of noise is detrimental for the sleep/wake rhythm, except for some special values for which it may favor regular oscillations. Depending on system parameters, noise induces regularities, such as stochastic resonance; if the natural light is turned off, it emerges that there is an optimal value of intensity noise which most deteriorates the regularity of the cycle; it is the phenomenon of anti-coherent resonance. Also, the phenomenon of stochastic resonance occurs; in the presence of natural light and there is an optimal noise intensity which improves the evolution of the wake/sleep system. However, there is a critical value of the noise beyond which the system becomes chaotic; indeed, for sufficiently high noise levels (how high depends upon the parameter of the system), the sleep/wake cycle evolves in a random and unpredictable manner, for whatever parameters of the external light.

Breathers on elliptic function background for a generalized nonlinear Schrödinger equation with higher-order terms

A generalized nonlinear Schrödinger equation with higher-order terms, which is derived as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin, is investigated. Firstly, in view of Riccati equations associated with the Lax pair, the Darboux transformation of a generalized nonlinear Schrödinger equation is presented. Secondly, the complicated Jacobi elliptic functions as seed solutions are considered so that much more fascinating solutions and dynamical properties can be obtained. Based on the above discussion, breathers in the presence of two kinds of Jacobian elliptic functions dn and cn are constructed. Finally, the dynamical properties of such solutions are analyzed by drawing the three-dimensional figures. The structures of these solutions are influenced by the higher-order operator. More importantly, the method provided in this paper can also be adopted to construct breathers on the elliptic functions background of other higher-order nonlinear integrable equations.

On Solving Fractional Higher-Order Equations via Artificial Neural Networks

On Solving Fractional Higher-Order Equations via Artificial Neural Networks

Multi-temperature generalized Zhdanov closure for scrape-off layer/edge applications

The derivation of the multi-temperature generalized Zhdanov closure is provided, starting from the most general form of the left-hand side of the moment-averaged kinetic equation with the Sonine–Hermite polynomial ansatz for an arbitrary number of moments. The process of arriving at the reduced higher-order moment equations, with its assumptions and approximations, is explicitly outlined. The generalized multi-species multi-temperature coefficients from our previous article are used to compute values of higher-order moments such as heat flux in terms of the lower-order moments. Transport coefficients and the friction and thermal forces for magnetic confinement fusion relevant cases with the generalized coefficients are compared to the scheme with the single-temperature coefficients previously provided by Zhdanov et al. It is found that the 21 N-moment multi-temperature coefficients are adequate for most cases relevant to fusion. Furthermore, the 21 N-moment scheme is also tested against the trace approximation to determine the range of validity of the trace approximation with respect to fusion-relevant plasmas. Possible refinements to the closure scheme are also illustrated to account for quantities which might be significant in certain schemes, such as the drift approximation.

Schwarz problem in a ring domain*

In this article, we discuss the solutions of Schwarz problem in a ring domain. Firstly we introduce the solutions of inhomogeneous Cauchy-Riemann equation. Afterwards we extend the discussions to higher-order equations. The relevant solvability conditions are obtained. The last part is reserved for the solutions of higher-order linear equations in .

On the Conditions for the Solvability of Boundary-Value Problems for Higher-Order Equations with Discontinuous Coefficients

On the Conditions for the Solvability of Boundary-Value Problems for Higher-Order Equations with Discontinuous Coefficients

Interactional solutions of the extended nonlinear Schrödinger equation with higher-order operators

In this paper, the extended nonlinear Schrödinger equation with higher-order operators, which can be diffusely used to describe the pulses propagating along an optical fibre, is under investigation. By means of the generalized Darboux transformation, we present the interactional solutions composed of the breather and rogue wave. Furthermore, regulating the coefficients of higher-order operators results in miscellaneous patterns of the interactions between the breather and rogue wave. Especially, the solutions in the extended nonlinear Schrödinger equation are more abundant than ones in the classical nonlinear Schrödinger equation. The kinetics of interactional solutions are elucidated graphically. The method provided in this paper can also be adopted to construct interactional solutions of other higher-order nonlinear integrable equations.

New approach to the singular solution of implicit ordinary differential equations

<p style='text-indent:20px;'>In this paper, for a class of nonlinear implicit ordinary differential equation with variable coefficients, a new approach to get the analytical singular solutions are given both for first order and higher order equations. Sufficient conditions on singular solutions are strengthened into necessary and sufficient conditions in specific case. Finally, the new results are verified by practical examples and MATLAB illustrations compared with traditional methods.</p>

Exact and numerical solutions of higher-order fractional partial differential equations: A new analytical method and some applications

In this paper, the solution methodology of higher-order linear fractional partial deferential equations (FPDEs) as mentioned in eqs (1) and (2) below in Caputo definition relies on a new analytical method which is called the Laplace-residual power series method (L-RPSM). The main idea of our proposed technique is to convert the original FPDE in Laplace space, and then apply the residual power series method (RPSM) by using the concept of limit to obtain the solution. Some interesting and important numerical test applications are given and discussed to illustrate the procedure of our method, and also to confirm that this method is simple, understandable and very fast for obtaining the exact and approximate solutions (ASs) of FPDEs compared with other methods such as RPSM, variational iteration method (VIM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM). The main advantage of the proposed method is its simplicity in computing the coefficients of terms of series solution by using only the concept of limit at infinity and not as the other well-known analytical method such as, RPSM that need to obtain the fractional derivative (FD) each time to determine the unknown coefficients in series solutions, and VIM, ADM, or HPM that need the integration operators which is difficult in fractional case.

Variable order step size method for solving orbital problems with periodic solutions

Existing variable order step size numerical techniques for solving a system of higher-order ordinary differential equations (ODEs) requires direct calculating the integration coefficients at each step change. In this study, a variable order step size is presented for direct solving higher-order orbital equations. The proposed algorithm calculates the integration coefficients only once at the beginning and, if necessary, once at the end. The accuracy of the numerical approximation is validated with well-known orbital differential equations. To reduce computational costs, we obtain the relationship for the predictor-corrector algorithm between integration coefficients of various orders. The efficiency of the proposed method is substantiated by the graphical representation of accuracy at the total evaluation steps.

On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

<p style='text-indent:20px;'>In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.</p>

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method; Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek; Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

Solvable three-dimensional system of higher-order nonlinear difference equations

In this work, we indicate three-dimensional system of difference equations xn = ayn?k + dyn?kxn?k?l/?bxn?k?l+?czn?l, yn = ?zn?k + ?zn?kyn?k?l/??yn?k?l + ??xn?l, zn = exn?k + hxn?kzn?k?l/?fzn?k?l +?g?yn?l , n ? N0, where k and l are positive integers, the parameters a,?b, ?c, d, ?, ??, ??, ?, e, ?f ,?g, h and the initial values x?j, y?j, z?j j = ?1,k+l, are non-zero real numbers, can be solved in closed form. In addition, we obtain explicit formulas for the well-defined solutions of the aforementioned system for the case l = 1. Also, the set of undefinable solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.

Bach-Einstein Gravitational Field Equations as a Perturbation of Einstein Gravitational Field Equations

The Bach equations are a version of higher-order gravitational field equations, exactly they are of fourth-order. In 4-dimensions the Bach-Einstein gravitational field equations are treated here as a perturbation of Einstein’s gravity. An approximate inversion formula is derived which admits a comparison of the two field theories. An application to these theories is given where the gravitational Lagrangian is expressed linearly in terms of R, R2, |Ric|2, where the Ricci tensor Ric = Rαβdxαdxβ is inserted in some formulas which are of geometrical or physical importance, such as; Raychaudhuri equation and Tolman’s formula.

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

A critical review on damage modeling and failure analysis of pin joints in fiber reinforced composite laminates

Fiber-reinforced composite material plays a vital role in structural engineering due to its lightweight and high strength ratio which becomes a key material in a mechanically fastened pin joint. Recent review articles in this area were restricted to the numerical and experimental approaches which are used for strength prediction of pin joints in polymer matrix composites. The present study begins with an extensive analysis of relevant studies in the provided structurally clamped joint region using numerous numerical approaches and theories of failure. Numerous experimental and numerical approaches are available nowadays and have been cited by the researchers in their respective research to foretell the damage initiation and failure mode in the composite joint. The study gives the review of different numerical analyses of composite joints by utilizing interactive criteria for failure analysis, viz. Tsai-Wu, Tsai-Hill, Yamada Sun’s theory predicts failure using higher-order polynomial equations that comprise all stress or strain components, whereas limit stress criteria i.e. Maximum stress criterion which uses linear equations for finding the solution. Progressive Damage Analysis (PDA) quantifies matrix and fiber failure using the material depletion rule preceded by Hashin’s theory which offers a good interpretation irrespective of the types of composite material. In the end, various parameters are discussed which enhance joint performance under different loading conditions.

A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier–Stokes differential equation transposes the latter into the Korteweg–De Vries–Burgers-equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.

A new algorithm based on compressed Legendre polynomials for solving boundary value problems

<abstract><p>In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.</p></abstract>