If it is released from rest, find the displacement of „y‟ at any distance „x‟ from one end at any time "t‟. The midpoint of the string is taken to the height „b‟ and then released from rest in  that position . ut (x,t) is then a function defined by (4) satisfying (1). Partial differential equations also began to be studied intensively, as their crucial role in mathematical physics became clear. Ordinary and Partial Differential Equations by John W. Cain and Angela M. Reynolds Department of Mathematics & Applied Mathematics Virginia Commonwealth University Richmond, Virginia, 23284 Nonlinear Differential Equations and Applications (NoDEA) provides a forum for research contributions on nonlinear differential equations motivated by application to applied sciences.. In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. In Mathematics, a partial differential equation is one of the types of differential equations, in which the equation contains unknown multi variables with their partial derivatives. Now putting x = 0 and x = 30 in (4), we have, ut (0,t) = u (0,t)  –us (0) =         40–40 = 0, and   ut (30,t) = u (30,t) –us (30) = 60–60 = 0, Hence the boundary conditions relative to the transient solution ut (x,t) are, and ut (x,0) = (4/3) x –20 ------------- (vi). If the temperature at Bis reduced to 0 o  C and kept so while that of A is maintained, find the temperature distribution in the rod. Solving this by the same method of separating variables, we have: = -ky                         :.=                                           y, :- y = e-kx+c = e-kx ec = Ae-kx  (where ec is a constant). C. Find the temperature distribution in the rod after time t. Hence the boundary conditions relative to the transient solution u, (4) A rod of length „l‟ has its ends A and B kept at 0, C respectively until steady state conditions prevail. wide and so long compared, to its width that it may be considered as an infinite plate. Since we are dealing with problems on vibrations of strings, „y‟ must be a periodic function of „x‟ and „t‟. The temperature of the end B is suddenly reduced to 60, C and kept so while the end A is raised to 40. (6)   A square plate is bounded by the lines x = 0, y = 0, x = 20 and y = 20. (1) Solve ¶u/ ¶t = a2 (¶2u / ¶x2) subject to the boundary conditions u(0,t) = 0, u(l,t) = 0, u(x,0) = x, 00, 0 £x £l. In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.. In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. To describe a wide variety of phenomena such as electrostatics, electrodynamics, fluid flow, elasticity or quantum, mechanics. This distinction usually makes PDEs much harder to solve than ODEs but here again there will be simple solution for linear problems. If a = 0 in our original equation (*), we get the first order equation of the same family. (9) A rectangular plate with insulated surface is 8 cm. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc. The objective of study of application of PDEs in Engineering is as follow; Any equation involving differentials or derivatives is called a differential equation. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is u(x,0) = 100 sin (. Find the displacement y(x,t). (diffusion equation) These are second-order differential equations, categorized according to the highest order derivative. If the temperature along short edge y = 0 is given. Of these three solutions, we have to select that particular solution which suits the physical nature of the problem and the given boundary conditions. The temperature along the upper horizontal edge is given by u(x,0) = x (20 –x), when 0, (9) A rectangular plate with insulated surface is 8 cm. We present examples where differential equations are widely applied to model natural phenomena, engineering systems and many other situations. An ode is an equation for a function of Whitham, G. B. Find the steady state temperature distribution at any point of the plate. Which can also be describe as an equation relating an unknown function (the dependent variable) of two or more variables with one or more of its partial derivatives with respect to these variables. Since „x‟ and „t‟ are independent variables, (2) can hold good only if each side is equal to a constant. Find the resulting temperature function u (x,t) taking x = 0 at A. Partial Differential Equations show up in almost all fields of exact sciences. The ends A and B of a rod 30cm. have the temperature at 30, A bar 100 cm. elliptic and, to a lesser extent, parabolic partial diﬀerential operators. The following faculty are especially active in the analysis of problems arising from PDEs. If a string of length ℓ is initially at rest in equilibrium position and each of its points is given the velocity, The displacement y(x,t) is given by the equation, Since the vibration of a string is periodic, therefore, the solution of (1) is of the form, y(x,t) = (Acoslx + Bsinlx)(Ccoslat + Dsinlat) ------------(2), y(x,t) = B sinlx(Ccoslat + Dsinlat) ------------ (3), 0 = Bsinlℓ   (Ccoslat+Dsinlat), for all  t ³0, which gives lℓ = np. Papers addressing new theoretical techniques, novel ideas, and new analysis tools are suitable topics for the journal. After payment, text the name of the project, email address and your The order of a differential equation is defined as the largest positive integer n for which an nth derivative occurs in the equation. Hence it is difficult to adjust these constants and functions so as to satisfy the given boundary conditions. u(x,0) = sin3(px/ a) ,0