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In this article, we study initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in { - 1 , 0 , 1 } {\{-1,0,1\}} . We construct an explicit formula for the weak solution. We also study the interaction of waves and the large time asymptotic behavior of a solution for the case when the initial data is periodic with zero mean over the period and also for the case when the initial data has compact support.

In this article, we study initial value problem for the zero-pressure gas dynamics system in non conservative form and the associated adhesion approximation. We use adhesion approximation and modi-ed adhesion approximation in the construction of weak asymptotic solution. First we prove a general existence result for the adhesion model for the initial velocity component in $$$H^s \mbox{ for } s$$$ > $$$ \frac{n}{2} + 1$$$ and the initial data for the density component being a $$$C^1$$$ function. Using this, we construct weak asymptotic solution for the system with initial velocity in $$$L^2 \cap L^{\infty}$$$ and the initial density being a bounded Borel measure. Then we make a detailed analysis of the explicit formula for the weak asymptotic solution and generalized solution for the plane-wave type initial data.

In this article we review some of the main contributions of Indian mathematicians in the theoretical analysis of partial differential equations in the last decade.

We construct solution of Riemann problem for a system of four conservation laws admitting $\delta$, $\delta'$ and $\delta''$-waves, using vanishing viscosity method. The system considered here is an extension of a system studied in [9] and [12] and admits more singular solutions. We extend the weak formulation of [12] to the present case. For the rarefaction case, the limit is not yet fully understood, the limit given in [12] is not correct and it does not satisfy the inviscid system. In fact we show that the limit of the third component contains $\delta$ measure and the fourth component contains the measure $\delta$ and its derivative, for a special Riemann data. We also solve Riemann type initial-boundary value problem in the quarter plane.

The 3-dimensional zero-pressure gas dynamics system appears in the modeling for the large scale structure formation in the universe. The aim of this paper is to construct spherically symmetric solutions to the system. The radial component of the velocity and density satisfy a simpler one dimensional problem. First we construct explicit solutions of this one dimensional case with initial and boundary conditions. Then we get special radial solutions with different behaviours at the origin.

In this paper, we study initial value problem for some non-conservative hyperbolic systems of partial differential equations of first order. The first one is the Riemann problem for a model in elastodynamics and the second o e the initial value problem for a system which is a generalization of the Hopf equation. The non-conservative products which appear in the equations do not make sense in the classical theory of distributions and are understood in the sense of Volpert (Math. USSR Sb. 2 (1967) 225). Following Lax (Comm. Pure Appl. Math. 10 (1957) 537) and Dal Maso et al. (J. Math. Pures Appl. 74 (1995) 483), we give an explicit solution for the Riemann problem for the elastodynamics equation. The coupled Hopf equation is studied using a generalization of the method of Hopf (Comm. Pure Appl. Math. 3 (1950) 201).

This paper is concerned with the boundary layers that arise in solutions of a nonlinear hyperbolic system of conservation laws in presence of vanishing diffusion. We consider self-similar solutions of the Riemann problem in a half-space, following a pioneering idea by Dafermos for the standard Riemann problem. The system is strictly hyperbolic but no assumption of genuine nonlinearity is made; moreover, the boundary is possibly characteristic, that is, the wave speed do not have a specific sign near the (stationary) boundary. First, we generalize a technique due to Tzavaras and show that the boundary Riemann problem with diffusion admits a family of continuous solutions that remain uniformly bounded in the total variation norm. Careful estimates are necessary to cope with waves that collapse at the boundary and generate the boundary layer. Second, we prove the convergence of these continuous solutions toward weak solutions of the Riemann problem when the diffusion parameter approaches zero. Following Dubois and LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible boundary traces. Following Part I of this work, we identify and rigorously analyze the boundary set associated with the zero-diffusion method. In particular, our analysis fully justifies the use of the scaling $1/\varepsilon$ near the boundary (where $\varepsilon$ is the diffusion parameter), even in the characteristic case as advocated in Part I by the authors.

In this paper we study a system of nonlinear partial differential equations which we write as a Burgers equation for matrix and use the Hopf-Cole transformation to linearize it. Using this method we solve initial value problem and initial boundary value problems for some systems of parabolic partial differential equations. Also we study an initial value problem for a system of nonlinear partial differential equations of first order which does not have solution in the standard distribution sense and construct an explicit solution in the algebra of generalized functions of Colombeau.

This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.

In this article, evolution of N-waves under the nonplanar Burgers equation, which takes into account geometrical expansion or contraction, is treated analytically. An exact asymptotic solution, generalizing that for the planar Burgers equation, is given for the case of expansion. An approximate treatement, using a balancing argument, gives asymptotic analytic results for both expansion and contraction. The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3].

This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L\infty bounds on the approximate solutions and so dealing with L\infty solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in ``Boundary conditions for nonlinear hyperbolic systems of conservation laws'', J. Diff. Equa. 71 (1988), 93--122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics., 43 pages

In this paper we show the existence of generalized solutions, in the sense of Colombeau, to a Cauchy problem for a nonconservative system of hyperbolic equations, which has applications in elastodynamics.

We derive explicit formulae for spherically symmetric solutions to the system of multidimensional zero-pressure gas dynamics and its adhesion approximation. The asymptotic behavior of the explicit solutions of the adhesion approximation is studied here. We observe that the radial components of the velocity and density satisfy a simpler equation, which enables us to get explicit formulae for different types of domains and study its asymptotic behavior. A class of solutions to the inviscid system with conditions on the mass instead of conditions at origin is also analyzed here.

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.

We consider the equation u t + u n u x + H ( x , t , u ) = 0 {u_t} + {u^n}{u_x} + H\left ( x, t, u \right ) = 0 and derive a transformation relating it to u t + u n u x = 0 {u_t} + {u^n}{u_x} = 0 . Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.

We extend a difference scheme introduced by Peter Lax for a scalar equation to a system of two conservation laws in the quarter plane {(x, t) : x > 0, t > 0} with initial conditions at t = 0 and conditions on the mass. We use Lax's transformation to linearize the problem and get an explicit formula for the solution. The limit generally belongs to the class of measures.

New diisocyanates, which contain tertiary nitrogen, have been prepared and used in the preparation of polyurethane cationomers. The ionomers were synthesized from commercial diisocyanates (toluene diisocyanate or hexamethylene diisocyanate) as well as from new diisocyanates [N,N′-bis(m- or p-isocyanato-phenyl)pyromellitic diimide] with polypropylene glycol, chain extended with 1,4-butanediol and/or N-methyldiethanolamine and then quaternized with 1,4-bis(chloromethyl)benzene. The cationomers thus formed were characterized by thermogravimetric analysis and differential scanning calorimetric measurements.

Polyurethane ionomers have been prepared using different diisocyanates (toluene diisocyanate and hexamethylene diisocyanate) and polyols (polyether and polyesters). These were chain extended with different chain extenders (N-methyl diethanol amine and 1,4-butanediol) followed by quaternization with dimethylsulfate to get the polyurethane cationomers. These ionomers were further dispersed in water to get aqeous polyurethane dispersions. These dispersions were sprayed on leather and physical properties of these leathers were studied.

Joseph, K.T., A Riemann problem whose viscosity solutions contain 8-measures, Asymptotic Analysis 7 (1993) 105-120. In this paper we study the Riemann problem for a 2 X 2 system of conservation laws, whose solution need not be real-valued functions, but may contain 8-measures concentrated along some curves. We consider a parabolic approximation of the Riemann problem and obtain an explicit formula. Then the solution of the Riemann problem is obtained as the limit of this approximate solution.

The influence of honey on biochemical and biophysical parameters of wounds was assessed by making an excision wound by cutting away a 4cm2 (2cm×2cm) full-thickness patch of skin from the shaven back of rats under ether anaesthesia. One milliliter of pure, commercial, unboiled honey was applied topically (Group II), or administered orally (Group III) or intraperitoneally (Group IV). The control rats were left untreated (Group I). The rate of healing was assessed by estimation of some biochemical parameters like DNA, total protein, total collagen, hexosamine and uronic acid contents of the granulation tissues formed different times after wound creation. All the biochemical parameters increased in honey-treated animals when compared with the control values. The increases were very significant with oral and intraperitoneal treatment when compared with those with the topical. Biophysical parameters like tensile strength, stress-strain behavior, rate of contraction, and period of epithelialization were also studied. There was an approximately 21% increase in tensile strength in Group II rats, whereas the increase was more in Group III (34.5%) and Group IV (52%) rats. The stress-strain behavior also followed the same trend. A significant increase in the rate of contraction was observed equally in all three honey-treated groups when their values were compared to the control. Among the three experimental groups, the intraperitoneal treatment decreased the period of epithelialization more significantly (37%) than the oral (22%) and topical (15%) treatments. The decrease in the period of epithelialization showed that the time taken by the wounds to heal was less with systemic treatment than with no treatment (control).

A method is presented for the design of a structure with specified low-order natural frequencies. The method can also be used for design problems in which certain bands, devoid of any frequency constraints, are specified. This can further be used to generate initial feasible designs for optimum design problems with frequency constraints.

In this paper we consider scalar convex conservation laws in one space variable in a stripD =(x, t): 0 ≤x ≤1,t > 0 and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip.

The changes in the activities of certain lysosomal hydrolases, viz., beta-glucuronidase, beta-N-acetylglucosaminidase, beta-galactosidase, beta-glucosidase, alpha-glucosidase, alpha-galactosidase, alpha-mannosidase, cathepsin B, cathepsin D, and collagenolytic cathepsin, in serum and heart of rats subject to myocardial infarction with isoproterenol, were studied during the periods of peak infarction and recovery. The activities of all the enzymes assayed exhibited a significant increase both in serum and in heart at peak infarction stage and these levels returned to normal during the stage of recovery and repair. The infiltration of inflammatory cells at the infarct regions and the altered lysosomal fragility are probably responsible for the increased activity of the enzymes studied. This may also bring about the catabolism of connective tissue constituents as reported in literature.

Following pioneering work by Fan and Slemrod who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid-vapor phase dynamics with {\sl physical} viscosity and capillarity effects taken into account. Following Dafermos we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis cover both the hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large Riemann data. The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic-elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity-viscosity ratio) are observed which prevent the solution to have globally bounded variation., Comment: 30 pages

In this paper we study the asymptotic behaviour of solutions of a system of $N$ partial differential equations. When $N = 1$ the equation reduces to the Burgers equation and was studied by Hopf. We consider both the inviscid and viscous case and show a new feature in the asymptotic behaviour., 9 pages

In this paper, we construct spherically symmetric solutions of the equations of compressible flow, which are important in the theory of explosion waves in air, water, and other media. Following McVittie [1], we write a general solution form, in terms of velocity potential,as a product of a function of time and a function of a similarity variable. First, we find solutions to the equations of motion and continuity without reference to adiabatic or isentropic relation. These solutions are quite general and can be applied to nonadiabatic motions,such as the motions of interstellar gas clouds that lose energy by radiation. All the solutions found by McVittie [ 1] have linear velocity profile with respect to distance. We introduce a nonlinear form of the velocity function containing an arbitrary function of the similarity variable. Adiabatic conditions lead to a second-order ODE,which we discuss in some detail. We relate our work to the earlier investigations of Taylor [ 2], McVittie [ 1], and Keller [ 3].

We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations., Comment: 7 pages

This is an erratum to [J. Appl. Anal. 15 (2009), no. 2, 247–267], http://dx.doi.org/10.1515/JAA.2009.247. In this note the author would like to make a correction in formulas (2.12) and (2.13) for the limit lim ϵ → 0 ( u ϵ ( x , t ) , v ϵ ( x , t ) ) ${\lim _{\epsilon \rightarrow 0}(u^\epsilon (x,t),v^\epsilon (x,t))}$ stated in Theorem (2.1); there is no change in the proof.

In this article we study Burgers equation and vector Burgers equation with initial and boundry conditions. First we consider the Burgers equation in the quarter plane x > 0, t > 0 with Riemann type of initial and boundary conditions and use the HOPf-cole transformation to linearize the problems and explicitily solve them. We study two limits, the small viscosity limit and the large time behaviour of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gredient of a scalar function. We investigate the asymptonic behaviour of this solution as time tends to infinity and generalize a rsult of HOPf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an intitial value problem etending the previous work of Sachdev et al. (1994). We also study the limits as viscosity parameter goes to 0. Finally, we get an explicit solution for boundry value problem in a cylinder.

Exact N-wave solutions for the generalized Burgers equation u(t) + u(n)u(x) + (j/2t + alpha) u + (beta + gamma/x) u(n+1) = delta/2u(xx),where j, alpha, beta, and gamma are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.

There are circumstances that make abdominal wall closure unsafe and technically impossible after laparotomy for trauma. In these difficult cases, prosthetic materials may be necessary to temporarily close the abdominal wall. To determine the optimal prosthetic in these instances, a retrospective chart review was conducted in our urban Level I trauma center. Twenty-five patients received 31 abdominal wall prostheses over a 4-year period. There were 7, 8, and 10 patients with 7 Marlex, 9 Dexon, and 15 Goretex prostheses, respectively. Each patient had only one type of prosthesis placed. The average age was 30.7 +/- 12.0 years, injury severity score was 20.3 +/- 7.4, and abdominal trauma index was 35.9 +/- 18.0; there was no significant difference in these values between groups. Eight patients died soon after the prosthesis was placed (average, 12.9 days) secondary to ongoing shock or multiple organ failure. Three of the seven surviving Goretex patients (43%) were intentionally left with small hernias. Three of the six Dexon patients (50%) were left with hernias; one of these eviscerated on day 150 and subsequently died, and the others have disabling gigantic hernias. Three of the four Marlex patients (75%) developed fistulae as a result of erosion into the small bowel or colon. One Marlex patient suffered with a chronically draining abdominal wound for 398 days prior to definitive closure. Goretex appears to be the best prosthetic for temporary abdominal wall closure because it causes less inflammatory reaction because of its smooth surface. It is therefore easier to retrieve at the time of definitive closure and carries less risk of fistula formation than other prostheses. Our Dexon patients suffered with gigantic hernias and one died because of complications of evisceration. We have abandoned the use of Marlex in abdominal wall closure because of the high incidence of fistula formation. We advocate the use of Goretex in temporary abdominal wall closure in this challenging group of patients.

An exact representation of N-wave solutions for the non-planar Burgers equation u t + uu x + ½ju/t = ½δu xx , j = m/n, m < 2n , where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for | x | < √(2 Q 0 t ), where Q 0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n , and are rather involved; explicit results are written out for j = 0, 1, ½, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be ‘singular’ and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are com pared with those of Crighton & Scott (1979).

The initial boundary value problem for the Burgers equation in the domain x greater-or-equal, slanted 0, t > 0 with flux boundary condition at x = 0 has been solved exactly. The behaviour of the solution as t tends to infinity is studied and the “asymptotic profile at infinity” is obtained. In addition, the uniqueness of the solution of the initial boundary value problem is proved and its inviscid limit as var epsilon → 0 is obtained.

The molecular organization of the skin collagen in the fish S. commersonianus has been investigated. The contents of imino acids proline and hydroxyproline are less in the skin collagen. The major collagenous component of fish skin is homologous to type I collagen. The number of CNBr peptides in fish skin collagen alpha 1 chains is two times that of rat skin alpha 1 CNBr peptides. The proline-hydroxyproline ratio in the six peptides studied was 1.66-3.5 as compared to that of rat skin collagen (0.67-1.94). This indicates that proline and hydroxyproline are not uniformly distributed in the collagen molecule in fish skin collagen.

The biochemical composition and molecular organization of myocommata collagen in the fish S. commersonianus has been investigated. The myocommata collagen was more acid soluble (89.3%). Glycine makes up close to one third of the amino acid residues in myocommata with proline and hydroxyproline being lower as compared to rat tail tendon collagen. In fish myocommata collagen two distinct chains, alpha 1 and alpha 2 were clearly detected along with their crosslinked components beta 11 and beta 12, typical of type I collagen.

The alterations in the heart lysosomal stability following isoproterenol induced myocardial infarction were studied in albino rats. The rate of release of beta-glucuronidase at various time intervals at 37 degrees C from lysosome rich fraction was taken as a measure of lysosomal stability. As compared to the control day one, three and five samples exhibited a significant increase in beta-glucuronidase activity at all the time intervals. The subcellular distribution of beta-glucuronidase was also studied and the soluble and total activities exhibited an increase at peak infarction stage and returned to normal during the recovery. The decrease in the lysosomal stability might be attributed to the increased beta-glucuronidase activity observed following myocardial infarction.

We consider the conservation law u t +[log(ae u +be −u )] x =0 where a>0, b>0, a+b=1, with initial condition u(x,0)=u o (x) in the quarter plane x>0, t>0. We are interested in studying the limit of the approximate solution defined by Lax's Difference Scheme and we discuss the problem of percribing u(0,t)=λ(t) arbitrarily

Two schemes are presented to generate the compatibility conditions for the integrated force method of discrete analysis. In the first scheme (recommended for analysis) the compatibility conditions are generated from the deformation-displacement relationship of the structure. This generation bypasses the popular concepts of redundants and basis determinate structure of the flexibility method. In the second scheme (recommended for design application), the compatibility conditions are obtained as a solution to a linear program (LP), in which the equilibrium equations are the constraints, and the linearized internal energy of the structure is the objective function. The LP is solved taking advantage of the sparsity of the coefficient matrix. The merits and limitations of the two schemes are compared to available procedures in the literature. As examples we take frame structures and plates.

Studies were carried out on grafting of various vinyl monomers to nitrocellulose by ceric ions. It was observed that graft copolymerization occurred only with methyl methacrylate (MMA) and methyl acrylate monomer. The variables such as initiator concentration, monomer concentration, time of grafting, and nitrocellulose content on grafting of MMA are discussed. By hydrolyzing away the nitrocellulose backbone, the grafted poly(methyl methacrylate) branches were isolated and the >c=o peak at 1740 cm −1 in the infrared spectra of these isolated branches gave definite evidence of grafting. The molecular weight of isolated branches has been determined by viscometry. The probable mechanism of grafting may be at the α-carbon atom of primary alcohol or at a C 2 -C 3 glycol group of the anhydro glucose unit or at the hemiacetal group of the end unit of nitrocellulose, as nitrocellulose is formed by the partial nitration of cotton cellulose.

This paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints. It ensures a non-singular basis matrix at every iteration. The same strategy can reliably be used within the generalized reduced gradient method for optimization under non-linear constraints. This method is applied to the minimum weight design of large structures under displacement and stress constraints, exploiting the sparsity of the constraint Jacobian matrix.

Poly(acrylonitrile-co-methyl methacrylate) films were oxidized in an air oven and in an oxidation chamber at temperatures of 80 to 200°C. Differential thermal analysis of these oxidized films showed that the exothermic peaks increased with increasing oxidation temperature. The change of viscosity, the insolubility in dimethylformamide and IR analyses indicate both intramolecular and intermolecular reactions.

SUMMARYIn three field experiments, peat pH increased in parabolic relation to liming at the rate of 0.0544 pH unit/t/ha between 0–40 t/ha lime, but only 0.0224/t/ha between 40–72 t/ha. Groundnut and sorghum seed yield and crop residue also showed parabolic responses to peat pH and liming, but groundnut seed yield correlated poorly with soil pH (r2 = 0.38, P = 0.05), compared with its crop residue as well as with sorghum seed and residue (r2 = 0.75, 0.76 and 0.79, respectively, P = 0.01). Groundnut was generally more tolerant of peat acidity (pH 3.35–3.65) than sorghum, giving higher plant survival in unlimed plots, optimum seed and residue at pH 4.60 compared with 5.70 for sorghum, and reduced responsiveness to pH increase. Seed and residue of both crops decreased with continuous cropping, probably because of increasing pest and disease incidence and depletion of available soil nutrients.