Order Partial Differential Equations Research Articles

On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a $ q $-th order PDE that admits a nontrivial Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order $ q-1. $ In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Channel flow dynamics of fractional viscoelastic nanofluids in molybdenum disulphide grease: A case study

Nanoscopic fluids especially viscoelastic Nanofluids are very useful in engineering and industrial problems. This research is to determine the open channel flow of a viscoelastic nanofluid (namely Oldroyd-B (OBNF)). Oldroyd-B fluid (OBF) was used as the base fluid and molybdenum disulphide nanopatrials were added in the fluid to form desired OBNF. To convert the mathematical model to a fractional model from a classical order partial differential equation PDE, fractional type derivative named Caputo-Fabrizio (CF) was used. The main objective of this study is to find the exact mathematical solution for the temperature, concentration and velocity distributions by using integral transformation technique. Final results are discussed graphically for the influence of different parameters on calculated temperature, concentration and velocity. Skin friction of the said fluid and engineering related dimensionless numbers including Reynolds number (Re), Prandtl number (Pr), Grashof number (Gr) and Schmidt number (Sc) are also discussed. At the end a comparison is illustrated in graphical form between current studied fluid (i.e. OBNF), another non-Newtonian fluid (i.e. Maxwell Nanofluid (MWNF)) and Newtonian fluid. As we know speed of Newtonian fluid is greater than non-Newtonian fluid, the same statement is validated by our solution. It is also noted that adding molybdenum disulphide nanoparticles to grease, heat transmission increased to 19.11% and mass transmission decreased to 2.51%.

Morphological Factors and Forces Acting in Normal and Sickled Erythrocytes in Sickle Cell Anemia: A Mathematical Model

We carried out a theoretical study that considers the morphological factors and forces acting on a normal and a sickled blood cell. Sickle cell disease is associated with vaso-occlusion which creates blood flow crises as a result of loss in shape memory of the normal red blood cell. Certain forces and chemical compositions acting on the cell could retain the shape memory. The problem of a second order partial differential equation modeled was solved to get the forces and chemical composition required to restore the sickled red blood cells back to its original shape. The results gotten showed that, increase in chemical reaction increased the RBC growth by creating a stretching force that causes the sickle cell to regain elasticity with improved oxygen.

Flexural Rigidity Influence on Dynamic Response of Orthoropic Rectangular Plate Resting on Constant Elastic Foundation

This research article considers the flexural rigidity influence along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions. The orthotropic rectangular plate model is a coupled fourth order partial differential equation having variables and singular coefficients. The solutions to this model are arrived at by reconstructing the fourth order partial differential. This partial differential equation model is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [11]. This set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to depict the influence of flexural rigidities along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions for both cases of moving distributed mass and moving distributed force.

Amplitude Variations of Moving Distributed Masses of Orthotropic Rectangular Plate with Elastically Supported Ends under Moving Loads

The deflection of thin orthotropic rectangular plate under moving loads is a classic problem in solid mechanics. However, the equations are challenging to solve due to their non linearity and complexity. At the same time, this equation is a coupled fourth order partial differential equation having variables and singular coefficients. In this research article, the partial differential equation is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [19]. This transformed set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble and Laplce transformation. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to solve the variations in amplitudes for some varying orthotropic plate parameters with elastically supported ends under moving loads for both cases of moving distributed force and moving distributed mass.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

Analytical solutions fractional order partial differential equations arising in fluid dynamics

Adaptive methods with [formula omitted] splines for multi-patch surfaces and shells

We introduce an adaptive isogeometric method for multi-patch surfaces and Kirchhoff–Love shell structures with hierarchical splines characterized by C1 continuity across patches. We extend the construction of smooth hierarchical splines from the multi-patch planar setting to analysis suitable G1 surfaces. The adaptive scheme to solve fourth order partial differential equations is presented in a general framework before showing its application for the numerical solution of the bilaplacian and the Kirchhoff–Love model problems. A selection of numerical examples illustrates the performance of hierarchical adaptivity on different multi-patch surface configurations.

The Double Inversion Technique in the Some Blaise Abbo Method Applied to Schrodinger-Type Fractional-Order Evolution Equations

In this article we have used the new double inversion technique in the Some Blaise Abbo(SBA) method to solve some fractional order partial differential equations (edp) in the Caputo sense of the Schrodinger type by integrating all boundary conditions.

A note on the telegraph type differential equation with time involution

In the present paper, the initial value problem for the telegraph type involutory in t second order linear partial differential equation with damping term is investigated. The equivalent initial value problem for the fourth order partial differential equations to the initial value problem for this second order linear partial differential equations with involution and damping term is obtained. Applying the operator tools, the stability estimates for the solution and its first and second order derivatives of this problem are established.

Impact of porous medium on natural convection heat transfer in plume generated due to the combined effects of heat source and aligned magnetic field

This study focuses on how the porous medium affects the plume generated due to line heat source when an aligned magnetic field is present. For this study, the momentum equation of the flow model is modified for porous medium by including the porosity term. A mathematical model is developed as coupled partial differential equations in order to study the flow problem. Later, a numerical solution is found for the system of coupled partial differential equations that are transmuted in to ordinary differential equations. For this purpose, the numerical characteristics of the problem are derived employing a shooting approach in combination with the built-in MATLAB tool bvp4c. The graphical illustrations of missing and specified boundary conditions demonstrate the impacts of porosity parameter [Formula: see text], magnetic force parameter S, Prandtl number [Formula: see text] and magnetic Prandtl number [Formula: see text] accompanied by a discussion of their corresponding physical implications. The novelty of this developed problem is proclaimed with justification by its emphasis on the principal characteristics of heat and fluid flow affected predominantly by the presence of a porous medium. The thorough examination of the porosity parameter [Formula: see text] for missing conditions depicts that the temperature and velocity profiles enhance while current density drops for the increasing values of the porosity parameter [Formula: see text]. Whereas, for specified conditions, the skin friction and magnetic flux enhance but heat transfer rate declines with increment in [Formula: see text].

Study of [formula omitted] dimensional fractional order non-linear Benney equation using an analytical technique

The area related to fractional order partial differential equations (FOPDEs) has also attracted attention. Therefore, we focuss our attention to study nonlinear Benney equations of using power law type Caputo derivative of fractional order. To compute a series-type solution for the problem under consideration, we combine the Adomian approach with the Laplace transform. Our findings are supported by graphical representations and numerical interpretations, illustrating the applicability and insights gained from this approach in solving complex real-world problems.

A problem with nonlocal integral 1st kind conditions for 4th order partial differential equation

A problem with nonlocal integral 1st kind conditions for 4th order partial differential equation

Planar Lorentz invariant velocities with a wave equation application

In this paper we determine the functional form of those planar velocity fields for which the associated system of two ordinary differential equations are automatically invariant under a Lorentz transformation. For planar motion we determine first order partial differential equations for the velocity components u(x,y,t) and w(x,y,t) in the x− and y−directions respectively and their general solutions in terms of two arbitrary functions. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz invariant energy–momentum relations and appear not to have been given previously in the literature. For a particular special relativistic model, one example is given involving similarity solutions of the wave equation. An interesting special case gives rise to a family of particle paths which are characterized by a single arbitrary function, and for which the magnitude of their velocities is the speed of light. This is indicative of the abundant possibilities existing in the “fast-lane”.

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

Inverse Coefficient Problem for the Coupled System of Fourth and Second Order Partial Differential Equations

Inverse Coefficient Problem for the Coupled System of Fourth and Second Order Partial Differential Equations

Functional gradation of material coefficients and size –effects in heat conduction: Numerical simulations

It is well known that the classical theory of heat conduction is scale invariant. On the other hand, it is experimentally evident that size-dependent effects are observable in small samples of micro/nano-scale dimensions. Incorporation of higher-order gradients of primary field variables into constitutive relationships yields a qualitative explanation of size-effects. Form the mathematical point of view, the governing equations are given by the partial differential equations (PDE) with higher-order derivatives in higher-grade continuum theories. As compared with classical theory of continua, the other complication of governing equations occurs in case of continuous media with functional gradation of material coefficients, when the problem is described by the PDE with variable coefficients. The traditional weak formulations are considered in global sense, hence the whole analysed domain/boundary is to be discretized into finite size elements. On the other hand, the strong formulations bring better computational efficiency because of elimination of integrations, but the price which should be paid is the need to approximate higher order derivatives of field variables. One of the most criticized shortcoming of the finite element (FE) approximation is its limited continuity on element intersections. The C0 continuity is insufficient for calculation of gradients of field variables on element intersections as well as for numerical solution of problems with governing equations of higher than 2nd order partial differential equations (PDE). Recently widely spread and elaborated mesh-free approximations utilize the higher order continuous shape functions. Another advantage is elimination of discretization of the analysed domain into finite elements, since only the nodes are scattered on the domain and its boundary. Both the strong and weak formulations are applicable in combination with mesh-free approximations. The Moving Finite Element Approximation (MFEA) and its utilization in mesh-free formulations for heat conduction problems is presented in this paper.

An immuno-epidemiological model with waning immunity after infection or vaccination

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies.

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables