All branches of physics present problems that can be traced to the integration of differential equations. More generally, the way of explaining all natural phenomena which depend on time is given by the theory of differential equations.”.
Preface to the Fourth Edition
Feynman We have made many additions and changes to modernize the content and improve the clarity of the previous edition. The index of the book has also been completely revised to cover a wide range of topics.
Preface to the Third Edition
Special emphasis is given to the fundamental similarities and differences in the properties of the solutions for the corresponding linear and non-linear equations under consideration. A new chapter on nonlinear PDEs of current interest and their applications has been added with considerable emphasis on the fundamental similarities and the distinguishing differences in the properties of the solutions to the nonlinear and corresponding linear equations.
Introduction
Brief Historical Comments
From the beginning of the study, considerable attention was paid to the geometric approach to the solution of differential equations. On the other hand, in the 1770s, Lagrange first began a systematic study of the first-order nonlinear partial differential equations in the form
Basic Concepts and Definitions
In the case of partial differential equations, the general solution depends on arbitrary functions rather than arbitrary constants. This is so because the general solution of a partial differential equation involves arbitrary functions;.
Mathematical Problems
If the solution is to have physical meaning, a small change in the original data must produce a small change in the solution. It is important to know that the process of making an approximation to the data produces only a small change in the solution.
Linear Operators
The principle of linear superposition is of fundamental importance in the study of partial differential equations. This principle is widely used in solving linear partial differential equations using the method of separation of variables.
Superposition Principle
It is also easy to verify that u(x, y; k) = e−kycos (k x), k ∈ R is a one-parameter family of solutions to the Laplace equation. In the case of an infinite number of terms in the linear combination (1.5.10), we require that the infinite set is uniformly convergent and sufficiently differentiable, and that all sets are Nk(ui), where N0=L,Nj=Mj forj= 1, 2,.
Exercises
First-Order, Quasi-Linear Equations and Method of Characteristics
Introduction
Classification of First-Order Equations
2.2.2) Similarly, the most general, first-order, partial differential equation in three independent variables ax,y,z can be written as. 2.2.7) Equation (2.2.4) is called a semilinear partial differential equation if its coefficients are and are independent of u, and thus, the semilinear equation can be expressed in the form
Construction of a First-Order Equation
It is clear that the general solution of a first-order partial differential equation depends on an arbitrary function. Its nature is similar to that of the singular solution of a first-order ordinary differential equation.
Geometrical Interpretation of a First-Order Equation
In fact, there are only two independent ordinary differential equations in the system (2.4.5); therefore the solutions consist of a two-parameter family of curves in (x, y, u) space. The projection on u= 0 of a characteristic curve on the (x,t) plane is called a characteristic base curve or simply characteristic.
Method of Characteristics and General Solutions
We now present an accurate formulation of the Cauchy problem for the first-order quasi-linear equation (2.5.1). Thus, the development of discontinuities between characteristics is an important feature of the solutions of partial differential equations.
Canonical Forms of First-Order Linear Equations
The second set, ξ(x, y) = constant, can be chosen to be any family of parameters of smooth curves that are nowhere tangent to the family of characteristic curves. In general, the canonical equation (2.6.7) can be easily integrated and the general solution (2.6.1) can be obtained after replacing ξ and η with the original variables maxandy.
Method of Separation of Variables
Usually the first-order partial differential equation can be solved by separating variables without the need for Fourier series. Since the left side of this equation is only a function of x and the right side is only a function of y, it follows that (2.7.2) can be true if both sides are equal to the same constant value λ which is called an arbitrary separation constant.
Exercises
Obtain the family of curves representing the general solution of the partial differential equation. Consider the Fokker-Planck equation (see Reif (1965)) in statistical mechanics to describe the evolution of the probability distribution function in the form.
Mathematical Models
Classical Equations
In this section, we list some of the more common linear partial differential equations that are important in applied mathematics, mathematical physics, and engineering.
The Vibrating String
Tsinβ−Tsinα=ρ δs utt (3.2.1) where ρ is the line density and δs is the minor arc length of the string.
The Vibrating Membrane
Since the deflection and slope are small, the area of the element is approximately equal to δxδy. The forces acting on the membrane element in the vertical direction are
Waves in an Elastic Medium
Inserting u=uT+uL into (3.4.18), taking into account the divergence of each term of the resulting equation, and then using (3.4.19a) gives Since the divergence of the term inside the square bracket is also zero, it follows that . 3.4.26).
Conduction of Heat in Solids
Since the rate of decrease of heat in D must be equal to the amount of heat leaving D per unit time, we have. If we assume that the integrand is not zero at a point (x0, y0, z0) in D, then, by continuity, the integrand is not zero in a small region surrounding the point (x0, y0, z0) .
The Gravitational Potential
If r represents the vector P Q, the force per unit mass at Q due to the mass at P can be written as. We assume that a particle with unit mass moves under the gravitational pull of the particle with mass m atP from infinity to Q.
Conservation Laws and The Burgers Equation
In many physical problems of interest, it would be a better approach to assume that q is a function of the density gradient ρx as well as ρ. Burgers (1948) first developed this equation to shed light on the study of turbulence described by the interaction of the two opposite effects of convection and diffusion.
The Schr¨ odinger and the Korteweg–de Vries Equations
Equation (3.7.9) occurs in many physical problems, including one-dimensional turbulence (where this equation originated), sound waves in a viscous medium, shock waves in a viscous medium, waves in fluid-filled viscous elastic tubes, and magnetohydrodynamic waves in a medium with finite electrical conductivity. More importantly, the properties of the solution of the parabolic equation are significantly different from those of the hyperbolic equation.
Exercises
Assuming that heat is also lost through radioactive exponential decay of the material in the rod, shows that the above equation becomes. Assuming that the material of the rod satisfies Hooke's law, show that the displacement function u(x, t) satisfies the general wave equation.
Classification of Second-Order Linear Equations
Second-Order Equations in Two Independent Variables
The classification of partial differential equations is suggested by the classification of the quadratic equation of conics in analytic geometry. The classification of second-order equations is based on the possibility of reducing equation (4.1.2) by coordinate transformation to a canonical or standard form in a point.
Canonical Forms
Consequently, the quadratic equation (4.2.4) has no real solutions, but it has two complex conjugate solutions which are continuous complex-valued functions of the real variables x and y. However, if the coecients A, B and C are analytic functions of x and y, one can consider equation (4.2.4) for complex x and y.
Equations with Constant Coefficients
Accordingly, c1 and c2 can take on complex values. 4.3.19) Applying this transformation easily reduces equation (4.3.5) to canonical form. Using these characteristic coordinates, we can convert the given equation into canonical form.
General Solutions
Summary and Further Simplification
Exercises
Use the polar coordinates randθ(x=rcosθ, y=rsinθ) to transform the Laplace equationnuxx+uyy= 0 into the polar form. Use the spherical polar coordinates (r, θ, φ) so that x=rsinφcosθ, y=rsinφsinθ,z=rcosφto transform the three-dimensional Laplace equationuxx+uyy+uzz= 0 into the form.
The Cauchy Problem and Wave Equations
The Cauchy Problem
The problem of determining a solution to equation (5.1.1) that satisfies the initial conditions (5.1.2) is known as the initial value problem. The Cauchy problem is now to determine the solution u(x, y) of equation (5.1.3) in the vicinity of the curve L0 that satisfies the Cauchy conditions.
The Cauchy–Kowalewskaya Theorem
It is then clear that the necessary condition for obtaining the second derivatives is that the curve L0 must not be a characteristic curve. If the coefficients of equation (5.1.3) and the function (5.1.5) are analytic, then all derivatives of higher orders can be calculated by the above process.
Homogeneous Wave Equations
Since the solution u(x, t) at each point (x, t) inside the triangular region Din, this figure is completely determined by the Cauchy data for the interval [x0−ct0, x0+ct0], the region Di is called the determinant region of the solution. We will now examine the physical meaning of the d'Alembert solution (5.3.8) in more detail. 5.3.12) Obviously, φ(x+ct) represents a progressive wave moving in the negative x-direction with speed c without change of shape.
Initial Boundary-Value Problems
At each point of the characteristic lines, the partial derivatives of the function u(x, t) do not exist, and therefore u can no longer be a solution to the Cauchy problem in the usual sense. The solution to the initial boundary value problem is therefore given by u(x, t) We note that for this solution to exist, f must be twice continuously differentiable and g must be continuously differentiable, and in addition.
![Figure 5.4.1 Displacement influenced by the initial data on [ x − ct, x + ct ].](https://thumb-ap.123doks.com/thumbv2/azdoknet/6348710.45829/151.661.151.519.127.437/figure-displacement-influenced-initial-data-x-ct-ct.webp)
Equations with Nonhomogeneous Boundary ConditionsConditions
The solution to the initial boundary value problem forx < ct is therefore given by. 5.5.4) Here dheg must satisfy the differentiability conditions, as in the case of the problem with homogeneous boundary conditions.
Vibration of Finite String with Fixed Ends
To observe the effect of boundaries on wave propagation, the features are drawn through the endpoint until they meet the boundaries and then continue inside, as shown in Figure 5.6.1.
Nonhomogeneous Wave Equations
Then the characteristics, x+y = constant, of equation (5.7.4) are two lines drawn through the point P0 with slope + 1. Let the sides of the triangle P0P1P2 be designated by B0, B1 and B2, and let D be the region representing the interior of the triangle and its boundaries B.
The Riemann Method
The solution on the right-hand side of the characteristic P1R1 is determined by the initial data given in Q1R2, while the solution. If the initial data on R1R2 has been changed, only the solution on the right side of P1R1 will be affected.
Solution of the Goursat Problem
When values are prescribed for both characteristics, the problem of finding a linear hyperbolic equation is called an initial value characteristic problem.
Spherical Wave Equation
In the context of fluid flow, represent the velocity potential so that the limiting aggregate flows through a center sphere at the origin and radius ris. 5.10.7). This represents the velocity potential of the point source, and ur is called the radial velocity.
Cylindrical Wave Equation
This is usually considered the cylindrical wave function due to a source of strength Q(t) at R= 0. Example 5.11.2. For a supersonic flow (M >1) past a fixed body of revolution, the perturbation potential satisfies the cylindrical wave equation.
Exercises
Show that it is a solution of the inhomogeneous diffusion equation with homogeneous boundary and initial data. Use Duhamel's principle to solve the inhomogeneous diffusion equation. n sinnx, is a solution to Laplace's equation.
Fourier Series and Integrals with Applications
- Introduction
- Piecewise Continuous Functions and Periodic FunctionsFunctions
- Systems of Orthogonal Functions
- Fourier Series
- Convergence of Fourier Series
- Examples and Applications of Fourier Series
In fact, the mean-square convergence theorem does not guarantee the convergence of the Fourier series for anyx. On the other hand, iff(x) is 2π-periodic and piecewise smooth onR, then the Fourier series (6.4.1) of the function f converges for everyxin−π≤x≤π.
