Motion of level sets by a generalized mean curvature

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΓ0 ｂｅａｓｍｏｏｔｈｃｏｎｎｅｃｔｅｄｈｙｐｅｒｓｕｒｆａｃｅａｎｄｉｔｉｓｔｈｅｂｏｕｎｄａｒｙｏｆａｂｏｕｎｄｅｄｏｐｅｎｓｅｔＵ Ｒｎ.Ｓｕｐｐｏｓｅｔｈａｔｔ0 ｉｓａｔｉｍｅ ,Ｍ =Ｍ(ｘ) ∈Ｃ3ｗｉｔｈＭ( 0 ) =0 ,Ｍ′(ｘ) >0 ,ａｎｄｌ1,ｌ2 ,… ,ｌｎａｒｅｐｏｓｉｔｉｖｅｒｅａｌｎｕｍｂｅｒｓ.{Γｔ} 0 ≤ｔ≤ｔ0 ｉｓａｃｌａｓｓｉｃａｌｓｍｏｏｔｈｓｕｒｆａｃｅ,ａｎｄｉｔｉｓｔｈｅｂｏｕｎｄａｒｙｏｆａｂｏｕｎｄｅｄｏｐｅｎｓｅｔＵｔ,ｗｈｉｃｈｉｓｄｉｆｆ…     

Fredholm and Compact Mappings Sharing a Value

1　ＰｒｅｌｉｍｉｎａｒｉｅｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＸ ,ＹａｎｄＺａｒｅＢａｎａｃｈｓｐａｃｅｓｏｖｅｒＫ =ＲｏｒＫ =Ｃ .ＯｎｅｍｅｔｈｏｄｔｏｓｏｌｖｅｔｈｅｅｑｕａｔｉｏｎＦ(ｘ) =0 ( 1 )ｉｓｔｈｅｃｏｎｔｉｎｕａｔｉｏｎｍｅｔｈｏｄ[1- 10 ].Ｔｈｉｓｍｅｔｈｏｄｃｏｎｓｉｓｔｓｏｆｅｍｂｅｄｄｉｎｇ ( 1 )ｉｎａｃｏｎｔｉｎｕｕｍｏｆｐｒｏｂｌｅｍｓＨ(ｔ,ｘ) =0 ,　　 0 ≤ｔ≤ 1 ,( 2 )ｗｈｅｒｅ ( 2 )ｉｓｅａｓｉｌｙｓｏｌｖｅｄｗｈｅｎｔ=0 .Ｉｆｉｔｉｓｐｏｓｓｉｂｌｅ…     

The Formation of Shock Waves of the Equations of Magnetohydrodynamics

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｎｅ＿ｄｉｍｅｎｓｉｏｎａｌＭＨＤｅｑｕａｔｉｏｎｓｕｎｄｅｒＬａｇｒａｎｇｅｃｏｏｒｄｉｎａｔｅａｒｅ τ ｔ- ｕ ｘ=0 , ｕ ｔ Ｐ ｘ 18π ｘ(Ｈ2 ｙ Ｈ2 ｚ) =0 , Ｈｙ ｔ Ｈｙτ ｕ ｘ =0 , Ｈｚ ｔ Ｈｚτ ｕ ｘ=0 , Ｓ ｔ=0 ,( 1 )ｗｈｅｒｅｕ ,τ,Ｈｙ,ＨｚａｎｄＳｔｈｅｖｅｌｏｃｉｔｙ ,ｓｐｅｃｉｆｉｃｖｏｌｕｍｅ,ｔｈｅｃｏｍｐｏｎｅｎｔｓｏｆｍａｇｎｅｔｉｃｆｉｅｌｄｉｎｙ＿ａｎｄｚ＿ｄｉｅｃｔｉｏｎｓａｎｄｔｈｅｅｎｔｒｏｐｙｒｅｓｐｅｃｔｉｖｅｌｙ .Ｔ…     

NGP_G-STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｉｎｇｔｈｅｓｔａｂｉｌｉｔｙｂｅｈａｖｉｏｕｒｉｎｔｈｅｎｕｍｅｒｉｃａｌｓｏｌｕｔｉｏｎｏｆｇｅｎｅｒａｌｉｚｅｄｎｅｕｔｒａｌｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｙ′(ｔ) =Ｌｙ(ｔ) Ｍｙ(ｔτ) Ｎｙ′(ｔτ)　　 (ｔ≥ 0 ) ,( 1 )ｙ(ｔ) =(ｔ)　　 (ｔ≤ 0 ) ,( 2 )ｗｈｅｒｅＬ ,ＭａｎｄＮ∈Ｃｄ×ｄａｒｅｃｏｎｓｔａｎｔｃｏｍｐｌｅｘｍａｔｒｉｃｅｓ,(ｔ) ∈Ｃｄｉｓａｇｉｖｅｎｖｅｃｔｏｒ＿ｖａｌｕｅｄｉｎｉｔｉａｌｆｕｎｃｔｉｏｎ ,ａ…     

BIFURCATIONS OF SUBHARMONIC SOLUTIONS IN PERIODIC PERTURBATION OF A HYPERBOLIC LIMIT CYCLE

Ｃｏｎｓｉｄｅｒｐｌａｎａｒｐｅｒｉｏｄｉｃｐｅｒｔｕｒｂｅｄｓｙｓｔｅｍ ｘ=ｆ(ｘ) +εｇ(ｔ,ｘ,ε ,δ) ,　　ｘ∈Ｒ2 ,( 1 )ｗｈｅｒｅε∈Ｒ ,δ∈Ｄ ＲｎｗｉｔｈＤｃｏｍｐａｃｔ,ａｎｄｆ,ｇａｒｅＣ3ｆｕｎｃｔｉｏｎｓａｎｄｇｉｓＴ＿ｐｅｒｉｏｄｉｃｉｎｔ,Ｔ >0 .Ｌｅｔｓｙｓｔｅｍ ( 1 )ｈａｓａｌｉｍｉｔｃｙｃｌｅＬ0 :ｘ =ｕ(ｔ) ,0 ≤ｔ≤Ｔ0 ｆｏｒε =0ｗｉｔｈＴ0 ｔｈｅｐｅｒｉｏｄｏｆＬ0 .ＳｕｐｐｏｓｅＴ0 /Ｔｉｓｒａｔｉｏｎａｌ,ｔｈａｔｉｓＴ0 /Ｔ=ｍ/ｋ,　　 (ｍ ,ｋ) =1 . ( 2 )Ａｓｗｅｋｎｏ…     

THE POSITIVE SOLUTION OF CLASSICAL GELFAND MODEL WITH COEFFICIENT THAT CHANGE SIGN

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒｔｗｏ＿ｐｏｉｎｔｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ (ＢＶＰ)(Ｐ) ｗ″(ｔ) +λｈ(ｔ)ｅｗ(ｔ) =0 ,　　 0 ≤ｔ≤ 1 ,ｗ( 0 ) =ｗ( 1 ) =0(λ>0 )ｈａｓｂｅｅｎｐｒｏｐｏｓｅｄｂｙＧｅｌｆａｎｄ[1]ｗｉｔｈｈ(ｔ) ≡ 1 ,ｔｈｉｓｉｓｔｈｅｒｅａｓｏｎｗｈｙｗｅｓａｙｔｈｅｐｒｏｂｌｅｍ (Ｐ)ｉｓａＧｅｌｆａｎｄｍｏｄｅｌ.Ｓｅｍｉｌｉｎｅａｒｐｒｏｂｌｅｍｓｏｆｔｈｉｓｔｙｐｅａｒｉｓｅｉｎａｖａｒｉｅｔｙｏｆｉｎｔｅｒｅｓｔｉｎｇａｐｐｌｉｃａｔｉｏ…     

A DAMAGE ACCUMULATING MODELING OF FAILURE WAVES IN GLASS UNDER HIGH VELOCITY IMPACT

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｉｌｕｒｅｗａｖｅｉｓａｓｐｅｃｉｆｉｃｆａｉｌｕｒｅｐｈｅｎｏｍｅｎｏｎｏｃｃｕｒｒｉｎｇｉｎｇｌａｓｓｍｅｄｉａｕｎｄｅｒｃｒｉｔｉｃａｌｉｍｐａｃｔｌｏａｄｉｎｇｗｉｔｈｈｉｇｈｓｔｒａｉｎｒａｔｅ (ｖ≥ 1 2ｋｍ/ｓ, ε≥ 1 0 5 /ｓ) .ＩｔｗａｓｆｉｒｓｔｏｂｓｅｒｖｅｄｉｎＫ1 9ｇｌａｓｓｐｌａｎａｒｉｍｐａｃｔｔｅｓｔｓｉｎ 1 991ｂｙＲａｓｏｒｅｎｏｖ[1]ｔｈａｔｔｈｅｒｅｅｘｉｓｔｓａｆａｉｌｕｒｅｌａｙｅｒｐｒｏｐａｇａｔｉｎｇａｔａｌｏｗｅｒｓｐｅ…     

Some theorems in the X-M-PN space

ＬｅｔＲｂｅｔｈｅｓｅｔｄｅｎｏｔｉｎｇａｌｌｒｅａｌｎｕｍｂｅｒｓ,Ｒ ｂｅｔｈｅｓｅｔｄｅｎｏｔｉｎｇａｌｌｎｏｎｎｅｇａｔｉｖｅｒｅａｌｎｕｍｂｅｒｓ.ＡｍａｐｐｉｎｇＦ:Ｒ→Ｒ ｉｓｃａｌｌｅｄａｄｉｓｔｒｉｂｕｔｉｏｎｆｕｎｃｔｉｏｎ,ｉｆｉｔｉｓａｎｏｎｄｅｃｒｅａｓｉｎｇｆｕｎｃｔｉｏｎａｎｄｌｅｆｔｃｏｎｔｉｎｕａｔｉｏｎｓａｔｉｓｆｙｉｎｇｔｈｅｆｏｌｌｏｗｉｎｇｃｏｎｄｉｔｉｏｎｓ:ｉｎｆｔ∈ＲＦ(ｔ)=0,ｓｕｐｔ∈ＲＦ(ｔ)=1,Ｄ ｔｈｅｓｅｔｏｆａｌｌｄｉｓｔｒｉｂｕｔｉｏｎｆｕｎｃｔ…     

Linearized oscillations for autonomous delay differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｏｎｌｉｎｅａｒａｕｔｏｎｏｍｏｕｓｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｘ′(ｔ) ∑ｍｉ=1ｐｉｆｉ(ｘ(ｔ-τｉ) ) =0 (1 )ａｎｄｔｈｅｌｉｎｅａｒｅｑｕａｔｉｏｎｘ′(ｔ) ∑ｍｉ=1ｐｉｘ(ｔ-τｉ) =0 ,(2 )ｗｈｅｒｅ ,ｐｉ ∈ (0 ,∞ ) ,τｉ ∈ [0 ,∞ )ａｎｄｆｉ ∈Ｃ(Ｒ ,Ｒ)ｆｏｒｉ =1 ,… ,ｍ .ＩｎＲｅｆ.[1 ] ,ｔｈｅｌｉｎｅａｒｉｚｅｄｏｓｃｉｌｌａｔｉｏｎｓｏｆＥｑ .(1 )ｗａｓｓｔｕｄｉｅｄａｎｄｉｔｗａｓｐｒｏｖｅｄｔｈａｔｉｆｔ…     

Singular perturbation of second-order nonlinear system with boundary perturbation

Ｗｅｃｏｎｓｉｄｅｒｉｎｔｈｉｓｐａｐｅｒｔｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｉｎｖｏｌｖｉｎｇｉｎｔｅｒｇｒａｌｏｐｅｒａｔｏｒεｙ″=ｆ(ｔ,ｙ,Ｔｙ,ε)ｙ′ ｇ(ｔ,ｙ,Ｔｙ,ε),(1)ｗｉｔｈｂｏｕｎｄａｒｙｐｅｒｔｕｒｂａｔｉｏｎｙ(ｔ,ε)｜ｔ=φ(ε)=α(ε),ｙ(ｔ,ε)｜ｔ=1 ψ(ε)=β(ε),(2)ｗｈｅｒｅε>0ｉｓａｓｍａｌｌｐａｒａｍｅｔｅｒ,ａｎｄφ(ε),ψ(ε)ａｒｅｂｏｔｈ,ｗｉｔｈｒｅｓｐｅｃｔｔｏε,ｓｕｆｆｉｃｉｅｎｔｌｙｓｍｏ…     

A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

ON CAUCHY PROBLEMS FOR THE RLW EQUATION IN TWO SPACE DIMENSIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｓｙｓｔｅｍｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅ ,ｔｈｅｐａｐｅｒｓ [1 ]ａｎｄ [2 ]ｈａｖｅｓｔｕｄｉｅｄｔｈｅｍｏｄｅｌｅｑｕａｔｉｏｎｏｆｓｉｍｐｌｅｗａｖｅｓｐｒｏｐａｇａｔｉｏｎｏｆｌｏｎｇｗａｖｅ ,ｕｔ+αｕｘ+ βｕｕｘ-γｕｘｘｔ=0　　ｘ∈Ｒ ,ｔ≥ 0ｉｓｃａｌｌｅｄｔｈｅＲＬＷｅｑｕａｔｉｏｎ .ＳＨＡＮＧ[3]ｏｂｔａｉｎｅｄｔｈｅｅｘａｃｔｓｏｌｉｔａｒｙｗａｖｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｔｗｏｄｉｍｅｎｓｉｏｎａｌＲＬＷｅｑｕａｔｉｏｎ .Ｉｎ…     

KINEMATIC WAVE PROPERTIES OF ANISOTROPIC DYNAMICS MODEL FOR TRAFFIC FLOW

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙｔｈｅａｕｔｈｏｒｓｐｒｅｓｅｎｔｅｄａｎｅｗｄｙｎａｍｉｃｓｍｏｄｅｌｆｏｒｔｒａｆｆｉｃｆｌｏｗ[1].Ｔｈｅｍｏｄｅｌｃｏｍｐｒｉｓｅｓａｃｏｎｔｉｎｕｕｍｅｑｕａｔｉｏｎａｎｄａｄｙｎａｍｉｃｓｅｑｕａｔｉｏｎ ｋ/ ｔ+ (ｋｕ) / ｘ=0 ,( 1 ) ｕ ｔ+ｕ ｕ ｘ=ｕｅ(ｋ) -ｕＴ +ａ ｕ ｘ,( 2 )ｗｈｅｒｅｋｉｓｔｒａｆｆｉｃｄｅｎｓｉｔｙ ;ｕｉｓｍｅａｎｓｐｅｅｄ ;ｘ ,ｔａｒｅｓｐａｃｅａｎｄｔｉｍｅｃｏｏｒｄｉｎａｔｅｓｒｅｓｐｅｃｔｉｖｅｌｙ .Ｔｉｓｒｅｌ…     

A class of two-level explicit difference schemes for solving three dimensional heat conduction equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｒｅｅ＿ｄｉｍｅｎｓｉｏｎａｌｈｅａｔｃｏｎｄｕｃｔｉｏｎｅｑｕａｔｉｏｎｉｎｔｈｅｒｅｇｉｏｎＤ :0≤ｘ,ｙ ,ｚ≤Ｌ ,0 ≤ｔ≤Ｔ ｕ ｔ= 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 ,ｕ｜ｘ=0 =ｆ1(ｙ,ｚ,ｔ) ,　ｕ｜ｘ=Ｌ =ｆ2 (ｙ ,ｚ,ｔ) ,ｕ｜ｙ=0 =ｇ1(ｚ,ｘ,ｔ) ,　ｕ｜ｙ=Ｌ =ｇ2 (ｚ,ｘ,ｔ) ,ｕ｜ｚ=0 =ｈ1(ｘ ,ｙ ,ｔ) ,　ｕ｜ｚ=Ｌ =ｈ2 (ｘ ,ｙ ,ｔ) ,ｕ｜ｔ=0 =φ(ｘ ,ｙ,ｚ) .(1 )(2 )…     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

ON NUMERICAL SOLUTIONS OF PERIODICALLY PERTURBED CONSERVATIVE SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｅｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓＬｕ≡ｕ″(ｔ) +ｇｒａｄＧ(ｕ) =ｐ(ｔ) ,ｕ( 0 ) =ｕ( 2π) ,ｕ′( 0 ) =ｕ′( 2π) ,( 1 )ｗｈｅｒｅｐ∈Ｃ(Ｒ ,Ｒｎ) ,Ｇ ∈Ｃ2 (Ｒｎ,Ｒ) .ＴｈｅｅｑｕａｔｉｏｎｍａｙｂｅｉｎｔｅｒｐｒｅｔｅｄａｓｔｈｅＮｅｗｔｏｎｉａｎｅｑｕａｔｉｏｎｏｆｍｏｔｉｏｎｏｆｔｈｅｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍｓｕｂｊｅｃｔｔｏｃｏｎｓｅｒｖａｔｉ…     

Traveling Wave Speed and Solution in Reaction-Diffusion Equation in One Dimension

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Periodic boundary value problems for first-order integro-differential equations of mixed type

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